Perhaps I’m being very ignorant here, but it seems to me in my ignorance that little attention is being given to a piece of critically important information the Scriptures give us:
“In the sixth month, the angel Gabriel was sent by God to a town in Galilee called Nazareth, to a virgin engaged to a man whose name was Joseph, of the house of David. The virgin’s name was Mary.” —Luke 1:26-27
The sixth month in the Jewish calendar is Elul, corresponding to August-September. A conception in Elul would mean a full-term birth nine months later in the month of Sivan, May-June.
Of course, we do not know that Mary conceived immediately after the visitation by Gabriel. It might have happened some time later.
Verse 5 tells us that Zachariah is from the line of Ajibah.
1 Chronicles 24 tells us that Ajibah (and thus Zachariahs) serving time in the temple was 8th.
All priests are required to serve during Passover and Shavu'ot so that adds 2 more weeks. That means Zachariah served in the temple the tenth week or the 2nd week of Sivan. After he served, Luke tells us he went home and Elizabeth got pregnant. Probably the 3rd week of Sivan.
Now Luke 1:24 says:
After these days Elizabeth his wife became pregnant, and she kept herself in seclusion for five months, saying, 25 “This is the way the Lord has dealt with me in the days when He looked with favor upon me, to take away my disgrace among men.”
5 months PLUS 11 weeks (almost 3 months)=8
Verse 26 says:
Now in the sixth month the angel Gabriel was sent from God to a city in Galilee called Nazareth, 27 to a virgin engaged to a man whose name was Joseph, of the descendants of David; and the virgin’s name was Mary.
Luke just got done talking about Elizabeths 5 months, and goes into talking about the 6 month. Luke isn't going back to the 6th month of the year, he's continuing the chronology of Elizabeth pregnancy. Which would probably be the 2nd or 3rd week of Kislev.
Verse 36:
36 And behold, even your relative Elizabeth has also conceived a son in her old age; and she who was called barren is now in her sixth month. 37 For nothing will be impossible with God.”
Again, Elizabeths 6th month.
Next verse 38:
And Mary said, “Behold, the bondslave of the Lord; may it be done to me according to your word.” And the angel departed from her. 39 Now at this time Mary arose and went in a hurry to the hill country, to a city of Judah, 40 and entered the house of Zacharias and greeted Elizabeth. 41 When Elizabeth heard Mary’s greeting, the baby leaped in her womb; and Elizabeth was filled with the Holy Spirit.
Some Bibles put a break between verse 38 and 39, which makes one think that time passed when Mary went to see Elizabeth. But if you take out the break, you see that the angel leaves and Mary, at THIS TIME arose and went in a hurry to the hill country...
Mary conceived right away.
I hope this helps.
John the Baptist was 6 months older then Jesus. So Mary conceived towards March. John the Baptist was conceived around September.