How do seasons work in Gen 5?

Question

Does the season change every month?

Answer 1

Yes, the seasons change the first day of every month, starting with Spring in January. Some Pokémon encounter rates change, overworld weather may change, music on certain routes change, and the day/night cycle will shift a few hours, depending on the season.

Source

Answer 2

Yes, the seasons change every month. January, May and September are Spring. February, June and October are Summer. March, July and November are Autumn. April, August and December are Winter.
Source: https://serebii.net/blackwhite/seasons.shtml
Hope I helped!