Right around Tabernacles. This is established by counting from the order of temple service of Zechariah, John the Baptist's father. This also accounts for the crowding at the inn, as pilgrims were on the way to/from the feast. That makes it Sept/Oct.
Independent verification was thought by the early Christians to reside in the fact that John says Jesus "dwelt among us, and we have seen His glory". This involves a parody on the Greek word for the feast of tabernacles, meaning "dwelt". The word is "skena", and matches the Hebrew work "shekinah", meaning the glory of God in visible form. John's inclusion of the phrase connecting the two words in thought to be a statement of the symbolism God would have placed in His birth.
There is a second position, that the Temple itself was in sin and following the wrong calendar. These folks use the Dead Sea Scrolls calendar. It is 364 days in a year, so in the 1400 or so years between Sinai and Jesus' birth, the calendars got six months out of sync. I do not accept this, as I do not believe that God would have failed to protect the Temple from error.