"Here’s what Luke tells us about Elizabeth’s pregnancy,
'After this his wife Elizabeth became pregnant and for five months remained in seclusion. ‘The Lord has done this for me,’ she said. ‘In these days he has shown his favor and taken away my disgrace among the people.’
'In the sixth month of Elizabeth’s pregnancy, God sent the angel Gabriel to Nazareth, a town in Galilee, to a virgin pledged to be married to a man named Joseph, a descendant of David. The virgin’s name was Mary' (Luke 1:24–27).
According to some biblical scholars, this allows us to pinpoint pretty accurately when Jesus would have been born. We start at the conception of John which would coincide with the service of the Abijah priests in June. Six months later, in December, Gabriel would make his appearance to Mary. Nine months out would put Christ’s birth in the Jewish month of Tishri—which would be September. The only problem with this theory is that it’s entirely dependent upon accurately knowing the month of Abijah’s service. Unfortunately, the Babylonian exile would have likely had an enormous impact on the service of Jewish priestly divisions—quite possibly causing a complete reset of those dates. We can’t entirely be sure when they would have served."
The site also mentions that Sheppards with their flocks would probably not have been out in the fields in late November-December as snow can occur in Jerusalem then. Also, a Roman census taking place throughout the empire would probably take place in more mild weather-so some scholars favor the time of September.
They have a wonderful excerpt, video, from the movie The Jesus Film on the site. The link is below.