Month of John's conception and birth
According to Parker and Dubberstein (Babylonian Chronology), Nissan 1 in 4 BC fell on March 29. According to Jewish custom, the first division of Jehoiarib served from Sabbath noon March 24 to Sabbath noon March 31 that year.
Then the second course served from March 31 through April 7. The third course started April 7, but as mentioned earlier, all the divisions officiated together during the Passover week. This caused the third course to end its administration on the Sabbath after the Passover week or the Days of Unleavened Bread, April 21. Then the weekly divisions resumed their regular order of service.
Continuing on, this would put the division of Abijah that year to serve from May 19 to May 26. Give or take a few days, we could approximate that John was conceived during the first week of June in 4 BC. And counting 9 months forward, he would have been born around March in 3 BC.
Month of Jesus' birth
And 6 months after the birth of John in March of 3 BC would place the birth of Jesus around September of 3 BC.
Sometime during the second week of September that year, would be Tishri 1 which is the Feast of Trumpets. Read Leviticus 23:24:
Lev 23:24 “Speak to the children of Israel, saying: ‘In the seventh month, on the first day of the month, you shall have a sabbath-rest, a memorial of blowing of trumpets, a holy convocation.
Since God uses His feast days to mark or symbolize important events, it is very likely that Jesus was born on the Feast of Trumpets.