My uneducated guess would be the Spring but the best guess would be when taxes were due that year. I believe that is why so many people were traveling or it could have been due to some other celebration that would explain why they were short of rooms at the inn.
Most likely to have been at the time of some great feast, such as Passover or Tabernacles. But as neither Matthew not Luke thought it imposrtant to tell us, I guess cistom is as good a way to choose when to celebate as any.