The scripture is silent with respect to the birth of the Messiah but gives us a powerful hint for the date of His conception.
In Haggai 2:7Haggai 2:7
English: World English Bible - WEB
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WP-Bible plugin the prophet gives us a messianic promise from YHWH. In verse 10 we see a date that is repeated four times in the chapter. In verse 4, we read His Name יְהֹושֻׁ֣עַ
This date is highly significant and can only be interpreted as the date that the Messiah was conceived in the womb of the virgin Miriam.
The date is the 24th of the ninth month aka Kislev. The importance of this date has been overshadowed with tradition celebrating the 25th of Kislev for the start of the Jewish feast of the re-dedication of the Temple after the Maccabees revolt.
If you count 9 months or 270 days after the 24th of the 9 month then you get to the 24th of the sixth month which is known as Elul (an acrostic for ani l’dodi l’dodi li = I am of my beloved and my beloved is mine). Which is the hypothetical date for the birthday of Yahushua. Note that on the 8th day He was presented in the Temple for His circumcision. The 8th day is Yom Teruah and that day would have been celebrated with Trumpets and this is a Day for the Coronation of Kings. Which was done with the sounds of Trumpets.
2Ki_9:13 Then they hasted, and took every man his garment, and put it under him on the top of the stairs, and blew with trumpets, saying, Jehu is king.
Psa_98:6 With trumpets and sound of cornet make a joyful noise before the LORD, the King.
It is unlikely that the birth happened on Yom Teruah because He would have been one week late. It is also unlikely that He was born on the pilgrimage for Sukkot because the Romans would not have called for a census during a pilgrimage feast when all the males were going to Jerusalem as per the commandment.