Possibility One:
The moon could be made of material whose chemical and/or physical structure gives it a different color when illuminated at different angles. If the moon is approximately spherical and one side faces the planet at all times, then sunlight will hit different regions of the planet at different angles at the same time.
Thus at full moon, when the moon was on the opposite side of the planet from the sun, the entire planet facing side would also be facing the sun, and it would all be illuminated by the sun. The moon would show concentric areas of different colors or shades, going from the center that was directly pointed at the sun to areas that were slanted away from the sun and had different shades or colors out to the outermost limb of the moon which would be tilted almost perpendicular to the sun.
At new moon, when the moon was passing lose by the sun during the daytime, it would be lit totally by planet light reflected from the planet. And the moon would be brightest at the point facing directly toward the planet, and get dimmer in regions farther away toward the limbs. Since the sunlight reflected from the planet would probably be much differ than the direct sunlight, the parts of the moon might lit by the reflected light might be too dim to show colors, and the sunlight reflected from the planet might have the color of the planet.
And when the moon was a little bit away from the new phase there would be a thin bright crescent on the side facing the sun, that would be much brighter than the rest of the moon and all one color.
And you can figure out how the phases in between new and full would look, I guess.
Of course, the moon will only look one single color when it is in a narrow crescent phase. When it is closer to full it should show concentric bands of different colors.
One solution might be that the atmosphere of the planet usually contains particles of light or dust that make it very hazy. If the moon appears smaller than Earth's moon as seen from the planet, and if the atmosphere is very hazy, the colors of the different sections of the moon might blend together in one color. When the moon is in a narrow crescent it might only reflect red light and appear red, when the moon is full it might reflect light of all colors from different sections which might be blended together by the atmosphere to look white (also the moon might look so bright when full that the colors can't be detected by the natives and it appears white), and in between it might look like some different color overall.
And maybe when the atmosphere is exceptionally clear telescopes might clearly show the different colored regions of the moon.
Possibility Two:
Long ago, super powerful aliens (or a previous fallen highly advanced civilization on that planet) colored the moon in broad bands.
They might have deposited colored materials in broad zones across the moon. Perhaps each zone went between meridians of longitude that were 30 degrees apart, making six zones across the near side of the moon: red, orange, yellow, green, blue, purple.
So as the moon goes through different phases the colors of the brightly lit regions of the moon would be:
1) Red.
2) Red & orange.
3) Red & orange & yellow.
4) Red & orange & yellow & green.
5) Red & orange & yellow & green & blue.
6) Red & orange & yellow & green & blue & purple.
7) Orange & yellow & green & blue & purple.
8) Yellow & green & blue & purple.
9) Green & blue & purple.
10) Blue & purple.
11) Purple.
Thus the moon of the planet will appear to change color with various phases. Of course it will only appear a single color when it is a very narrow crescent. When it is fuller it will show bands of color.
If the moon has a much smaller apparent diameter when seen from the planet than the Moon has from Earth, the bands of color might seem to blend together as seen from the planet. And if the atmosphere of the planet is usually very hazy it may usually blend the colors of the different regions of the moon together.
So as the moon goes through various phases it may seem red at first crescent form when only the red region is illuminated; orange when half full and the red, orange, and yellow regions are illuminated; white when it is full and all regions are illuminated and all the colors blend together to make white; blue when half full and and the green, blue, and purple regions are illuminated, and purple when crescent and only the purple region is illuminated.
Of course if the bands are laid out and colored differently the moon might have different colors in its different phases.
And maybe when the atmosphere is exceptionally clear telescopes might clearly show the different colored regions of the moon.