On a planet, seasons are dictated by the axial tilt. The colder seasons located in the hemisphere furthest from the sun, and the warmer seasons in the hemisphere closest to sun. Temperatures, and hence the seasons, in the different latitudes are determined by the angle of the sunrays hitting the land. If your moon had an axial tilt, this would be affected it in a similar manner.
However, your moon is also orbiting around the planet, moving further away from the sun and then closer to the sun and then back around the planet again. This orbit-change in distance from the sun would be much larger than the hemispheric-change in distance from the sun, and would affect both hemispheres of the moon at the same time. It would affect the overall climate and not just the seasons. Ie say it was what we would consider the "summer" hemisphere on the moon. It could be colder in that summer-hemisphere during the darker periods of that month when furtherest away from the sun, than compared to the winter-hemisphere during the bright days on the nearest sun-side of the planet.
This monthly moon orbit would be similar to both the more long-term Annual and even longer-term Milankovich Cycles which influences Earth's overall climate. The planet being closer and further away from the sun, aphelion and perihelion. The Milankovich climate cycle is thousands of years long and influences the development of ice-ages etc. Your moon would be experiencing the equivalent of a very much smaller-scale "Milankovich Junior" Cycle every lunar month, getting closer and further away from the sun as it orbited your planet. You would then also still have both the planet's Annual and Milankovich Cycles as well.
If you combine this moon orbit (climate) with a moon axial tilt (seasons) then you can have;
- "winter" months with cooler weeks (sun-side) and very very cold dark weeks (far-side).
- "summer" months with warmer weeks (sun-side) and still very cold dark weeks (far-side).
- The moon's orbit around the planet would have much larger effect on your lunar climate than the seasonal axial tilt.
So now you just need to figure out if you want your moon to take 30 days to orbit your planet or 90 days!