There have been other questions about habitable moons, and you should read them for other information about hypothetical habitable moons.
This one, for example:
Characteristics of a habitable satellite planet1
I have answered enough such questions that I have got in the habit of referring to my previous answers.
My answer to this question:
What should the size of my moons be? 2
points out that its is considered impossible for a moon to have a month that is the same length as the year of the planet it orbits. A moon's orbit wouldn't be stable unless it orbited the planet at least 9 times during one orbit of the planet around their sun.
If a moon orbits a planet, it should have either a normal rotation period or else have a locked rotation period, so that one side of the moon would always face the planet and one side would always face away from the planet.
If the moon has a normal rotation period, the planet it orbits will be visible for half of every day from almost every part of the moon.
If the moon is tidally locked, the natives of the outer or far side of the moon will never see the planet in their skies and will never directly observe it from their side of the moon. They can only hear about it from natives of the near side or see it themselves when they explore the near side of their moon.
This reminds me of the James Blish story "Get Out of My Sky" (1960) and the Poul Anderson story "The Longest Voyage" (1960).
Because of atmospheric refraction of light and libration effects the planet will be visible from some parts of the far side that are close to the line between the far and near sides.
So your far side natives should be limited to only part of the far side of their moon so they will never see their primary planet.
One suggestion would be that there should be a giant impact feature on the far side of the moon from the planet with alternating rings of flat plains and high ring wall mountains. There can be a central mountain range surrounded by higher and dry plains surrounded by lower plains covered by water in a ring shaped ocean surrounded by a circle of land with high mountains in the spine of the circle of land, surrounded by a ring shaped ocean surrounded by a ring of land, and so on.
The central body of land should be like a small continent in size, large enough for a large civilization to arise on the central continent and on the flatter parts of the ring shaped continent of land beyond the ring shaped ocean around the central continent.
The immediate ring shaped continent beyond the ring shaped ocean should have a ring shaped spine of central mountains running all the way around, and they should be high enough for most of the passes to be covered in glaciers all year round. So almost nobody has ever crossed those mountains from one side of the mountains to the other side. As far as people on the inner side of the ring shaped continent - and those on the innermost continent - know the ring shaped glaciated mountain chain could be at the edge of a presumably flat, disc shaped world. They might believe the gods built the ring mountains to keep air and water from falling off the edge of the world.
And possibly any farther out ring shaped continents may also have mountain rings tall enough to be glaciated all year and impassable.
And so the civilizations at the center of the giant impact feature might never have heard anything about the giant planet that is always visible on the other side of their moon.
As said above there should be at least nine month/days when the moon orbits the planet for each year of the planet as it orbits around the sun and possibly tens or hundreds of month/days per year.
The gas giant planet the habitable moon orbits should have an axial tilt, and tidal forces will have regularized the habitable moon's orbit so that the moon's axial tilt will be almost exactly the same as the planet's and so that the moon's orbital plane will be almost exactly in the equatorial plane of the planet.
So the moon will share the axial tilt of the planet, and the lower the axial tilt is the less noticeable the seasons will be on the habitable moon, and the higher the axial tilt the more noticeable the seasons will be on the habitable moon.
So the natives of the habitable moon should notice seasons which are better or worse to some degree for hunting, fishing, food gathering, planting, and harvesting cops. And thus they will keep track of time and develop calendars and observational astronomy to keep track of and predict the passage of time and the seasons.
And the natives will also keep track of their days and nights, which of course will be each be about one half of an orbital period around the primary planet.
The natives on the near side will see the gas giant planet and may assume for many thousands of years that it - along with their sun and other planets - orbits their moon before they eventually become advanced enough to realize that all the planets orbit their sun and that they are on a moon orbiting the gas giant planet.
And the natives of the far side of the moon wouldn't see the planet or know it was there, but will eventually be able to discover that the planets in their solar system orbit around their sun, and that their moon seems to be one of those planets. And then they might discover that there are so many problems with the orbit of their "planet" that making it a moon orbiting a planet that can't be seen from their side is the simplest explanation.
As we all know, every night at midnight the stars that are on a line across the sky from north to south are at the opposite side of Earth - and/or of the hypothetical celestial sphere that for thousands of years they were supposed to be attached to - from the Sun.
On Earth a sidereal year is the time it takes the Earth to make a complete orbit of the Sun as measured against the stars. It is 365.256 days. In an average sidereal year the Earth travels about 1.0146 degrees along its orbit around the Sun. A stellar day is the time it takes Earth to rotate 360 degrees with respect to the stars.
So each day at midnight the stars appear to have moved about 1.0146 degrees from their positions the previous midnight, and over the course of a year the midnight line will appear to move 360 degrees around the celestial sphere to its original position.
But the natives will originally discover what are called tropical years and solar days. Because the Earth moves along its orbit during a sidereal day, at the end of a sidereal day the direction that used to point to the Sun is now pointing about 1.0146 degrees away from the Sun. A stellar day is the time period when the Earth turns 360 degrees relative to the Sun.
And a tropical year is the time period for a complete cycle of the seasons, and is about 365.242 days long.
Suppose that it took exactly 450 month/days of the habitable moon for the planet to orbit its sun. Each month/day the midnight line would point 0.8 degrees off of where it pointed the previous midnight.
Suppose that it took exactly 90 month/days of the habitable moon for the planet to orbit its sun. Each month/day the midnight line would point 4 degrees off of where it pointed the previous midnight.
Suppose that it took exactly 9 month/days of the habitable moon for the planet to orbit its sun. That is about the least possible number of month/days for the planet to orbit its sun. Each month/day the midnight line would point 40 degrees off of where it pointed the previous midnight.
Of course the year of the planet wouldn't be evenly divisible by the month/day of the habitable moon.
Obviously the fewer month/days of the habitable moon there are in a year of the giant planet, the more noticeable will be the differences between sidereal and tropical years, and between stellar days and solar days. And the more month/days of the habitable moon that there are in a year of the giant planet, the less noticeable will those differences be.
In our solar system, and in any solar system like ours, the distances between the planetary orbits will be so great that every planet will look like a dot of light when seen with the naked eye from another planet even at their closest approaches. But when telescopes are invented (first used for astronomical observations from Earth in 1609) and used for astronomical observations some of the planets should show discs in the telescopic views, and thus their phases should be observable.
The differences between the phase cycles of inner and outer planets should provide strong evidence in favor of a theory that the planets orbit around their stars.
All four of the Galilean moons of Jupiter are bright enough to theoretically be seen with the naked eye from Earth. When Jupiter and Earth are closest, their apparent magnitudes range from 4.6 to 5.6. But their angular separation from Jupiter never gets to be more than about the absolute minimum angle that the human eye can see, so they appear as part of the same dot of light as Jupiter.
Even cheap binoculars of the present time are superior to the early telescopes which discovered the Galilean moons of Jupiter that were discovered in December, 1609 or January, 1610. The discovery of the Galilean satellites clearly seen to orbit Jupiter showed that astronomical objects could orbit around other astronomical objects that were not the earth, and was a strong argument in favor of the heliocentric theory.
Since your fictional star system is different from our solar system in some ways - since it has a gas giant planet with a giant habitable moon orbiting in its habitable zone - it could be different from our solar system in other ways, including the relative and absolute spacing of the planets.
Many exoplanets and systems of exoplanets have been discovered, so it is known that the majority of star systems are greatly different from ours in various ways.
For example, CVSO 30 has the widest spacing, in both absolute and relative terms, between two consecutive (known) planets of a star. CVSO 30 c is about 78,998 times as far from their star CVSO 30 as CVSO 30 b is, or about 662 Astronomical Units (or AU) - an AU is the distance from Earth to the Sun.
If the nearest planet to your habitable moon and its giant planet orbits hundreds of AU farther away from their star, it may appear like a mere dot of light in early telescopes, and later and better telescopes, and still later and better telescopes, and so on. It might not be seen as a disc with phases until 20th century telescopes or 21st century telescopes are invented.
And similarly it may not be possible to see moons orbiting such a distant planet, supporting a theory that the planets orbit around their star, until 20th century or 21st century telescopes are invented.
And on the other hand, in some star systems exoplanets orbit many times closer together than any planets in our solar system.
The smallest absolute difference between the orbits of two consecutive planets is between Kepler-70b and Kepler-70c. It is 0.0016 AU or about 240,000 kilometers.
During closest approach, Kepler-70c would appear 5 times the size of the Moon in Kepler-70b's sky.
Note that there are reports of an unconfirmed planet orbiting between the orbits of Kepler-70b and Kepler-70c!
The Kepler-36 system has the smallest known relative difference between the orbits of two consecutive planets. Kepler-36c is believed to have an orbit only 11 percent wider than that of Kepler-36b.
Kepler-36b and c have semi-major axes of 0.1153 AU and 0.1283 AU respectively, c is 11% further from star than b.
The potentially habitable planets in the habitable zone of TRAPPIST-1 also orbit quite close to each other.
The system is very flat and compact. All seven of TRAPPIST-1's planets orbit much closer than Mercury orbits the Sun. Except for TRAPPIST-1b, they orbit farther than the Galilean satellites do around Jupiter, but closer than most of the other moons of Jupiter. The distance between the orbits of TRAPPIST-1b and TRAPPIST-1c is only 1.6 times the distance between the Earth and the Moon. The planets should appear prominently in each other's skies, in some cases appearing several times larger than the Moon appears from Earth. A year on the closest planet passes in only 1.5 Earth days, while the seventh planet's year passes in only 18.8 days.
Thus it is possible for some solar systems to have planets so close that they sometimes or always have visible discs as seen with the naked eye from the surfaces of some or all other planets in that system.
If planets are close enough to show visible discs with the naked eye, their phases can be seen with the naked eye, and the naked eye can tell the difference between the phases of inner planets and outer planets, thus forming strong evidence for a theory that the planets orbit around their star instead of the star and planets orbiting round the habitable moon.
In our solar system, some people allegedly have seen phases of Venus with the naked eye:
The extreme crescent phase of Venus can be seen without a telescope by those with exceptionally acute eyesight, at the limit of human perception. The angular resolution of the naked eye is about 1 minute of arc. The apparent disk of Venus' extreme crescent measures between 60.2 and 66 seconds of arc,4 depending on the distance from Earth. Nevertheless it is possible for observers with extremely acute eyesight to see a crescent Venus under ideal atmospheric circumstances.
There have been numerous reports stating such observations. The phases of Venus are alleged to have been seen in Mesopotamian times by priest-astronomers. Ishtar (Venus) is described in cuneiform text as having horns.1 However, other Mesopotamian deities were depicted with horns, so the phrase could have been simply a symbol of divinity.
So in a solar system where planets were a little closer together than in ours people might be able to see the crescent phase of the next innermost planet with the naked eye. And if the planets were much closer together they might be able to see all the phases of that planet with the naked eye.
And if in some solar systems planets can get close enough to show visible discs and phases to the naked eye, in some solar systems planets might get close enough to see moons orbiting those planets with the naked eye, which would be a strong argument in favor of a theory that the planets orbit their star.
It may be noted that it is possible that some humans have seen one or more of the moons of Jupiter with the naked eye.
Clearly if Jupiter could get half as far, or a quarter as far, as it actually gets, it might be possible to see the Galilean satellites regularly enough with the naked eye to plot their obits around Jupiter. If Jupiter could get much closer than that it would be even easier to see the orbits of the Galilean moons.
So the structure of your fictional star system will determine how advanced the natives of the far side of your moon have to be in order to discover that their world orbits around a point in space that orbits around their star, and that there should be an astronomical body at that point in space, a point in space and astronomical body that their side of the moon is always turned away from.