"There was in the days of Herod, the king of Judea, a certain priest named Zechariah, of the division of Abijah. His wife was of the daughters of Aaron, and her name was Elizabeth."
Luke 1:5
Zechariah, the father of John the Baptist, is assigned in the division of Abijah... the schedule of service of Abijah in OT is in the eighth (I Chronicles 24:6-19) which is in the month of Tammuz (first part is of Hakkoz, second part is of Abijah)... Tammuz is mid-June to mid-July...
so after Zechariah received the news from angel Gabriel, and after his service, he went home (Luke 1:23), which is mid-July...
Elizabeth, who is 6-month pregnant with John (January), marks Mary being conceived by the Holy Spirit with Jesus' human form... so this marks John's birthmonth around April, and Jesus' birthmonth around October... that is, considering both mothers (Elizabeth and Mary) delivering a normal 9-month pregnancy...
Jesus' birth will never be winter... it's not logical for Augustus Caesar to decree a census for all people under his authority to register in the time of winter (Luke 2:1-2)... and if it's winter, there should be no country shepherds living out in the fields, keeping watch over their flock by night (Luke 2:8)...
in Jerusalem,
winter is December to March
spring is March to May
summer is June to September
autumn is September to November