His priestly division served two rotations each year. The second rotation would have fallen in October near the Day of Atonement.
The weight of tradition would suggest that the account in Luke takes place during that second rotation.
Discussion
I don't know. What i read is that Zacharia's on call duty was that of Avia which was the 8th and thus in the month Tammuz. Also that there were sepherds in the countryside when Jesus was born and that sepherds are in the countrydide from March until October. Can you have a look on this information?
In fact from a religious perspective i don't care, it is irrelevant. But it seems interesting. So i am curious (as always)