Luke 1:26 - In the sixth month the angel Gabriel was sent from God to a town of Galilee call Nazareth to a virgin betrothed to a man named Joseph...
Ancient Christianity had a tradition that Jesus died on the day he was conceived. In Judaism, the sixth month was/is Adar, which falls within our months of February, March. The traditional dates of the annunciation/crucifixion is March 25.
Birthdays were not commonly noted/celebrated in those days. Once celebrating birthdays did become popular, Christians counted nine months from March 25...December 25.
Luke wrote his Gospel/Acts in Greek. Ancient Greek calendar would have placed the sixth month in December. However, Luke lived in Roman times and their sixth month was August. (August literally means sixth).
Early Christians lived in Roman times, but because of their celebration of the Annunciation being in March, it adds some weight to Luke also using the Jewish calendar.
On the other hand (there is always an 'other hand', Luke's Gospel was written most likely by a non-Palestinian to an audience of largely Gentile Christians.
Interesting side note is the Jewish celebration of Purim is in their sixth month, a celebration where people give gifts to each other and to the poor.