Speculation abounds regarding the year and the month in which Jesus was born. Although most views range from 6 to 4 B.C., there are some suggestions that he could have been born in 7 or even 8 B.C. The Bible provides only a few clues.
Luke’s gospel sets the scene by saying that after Mary was visited by the angel Gabriel, Mary went to see her cousin Elizabeth who was six months pregnant (Luke 1:36). Mary stayed with Elizabeth for about three months before she returned home (Luke 1:56). Below is a comment from the NIV Bible regarding Matthew 3:1:
John the Baptist, the forerunner of Jesus, born circa 7 B.C. to Zechariah (a priest) and Elizabeth (cousin to Mary).
It is worth remembering that in Judaism the age of a child is counted from its conception, not its birth. If Jesus was conceived during Chanukkah – Festival of Lights (December, when Elizabeth was 6 months pregnant), he would have been born during the Festival of Sukkot (end September or beginning October) 7 B.C.
Luke’s gospel says that Jesus’ birth happened during the time when Quirinius was governor of Syria. “In those days Caesar Augustus issued a decree that a census should be taken of the entire Roman world. (This was the first census that took place while Quirinius was governor of Syria)” (Luke 2:1–2). This is problematic because history suggests Quirinius was governor between A.D. 6 and 7, about 10 years after the birth of Jesus. Here is one explanation:
The Greek word for “first” in Luke 2:2 is a form of the word protos and can be translated “before.” Thus Luke 2:2 could actually be translated, “This was the census taken before Quirinius was governor of Syria.” Quirinius actually ruled Syria on two separate occasions, and there were actually two censuses taken. The “first census” mentioned in Luke 2:2 occurred during Quirinius’ first term as governor, and another during his second term. The second census is mentioned in Acts 5:37 and probably took place between A.D .6 and 7 (Josephus links this census to an uprising led by Judas of Galilee). Luke was the author of both the Gospel of Luke and the book of Acts, and his goal was to write “an orderly account” (Luke 1:3). It seems that Luke did write a careful, orderly account: he mentions two censuses, and it was during the first one that Jesus was born. It would be unlikely for such a meticulous historian to make a blatant mistake in his timeline of events. Source: https://www.gotquestions.org/Quirinius-census.html
With regard to the census mentioned in Luke 2:1-2, 5th century historian Orosius records that Augustus “ordered a census of each province everywhere and all men to be enrolled.” Josephus notes that, "When all the people of the Jews gave assurance of their goodwill to Caesar, and to the king's government, these very men [the Pharisees] did not swear, being about six thousand." This first census took place before the death of Herod the Great in 4 B.C. There was also a census in 8 B.C. but for Roman citizens only.
Another school of thought gives the following dates:
The conception of John 23rd Sivan = June 23-24, 5 B.C.
The birth of John 7th Nisan = March 28-29, 4 B.C.
The Miraculous "Begetting" 1 Tebeth = December 25, 5 B.C.
The Nativity 15th Tisri = September 29, 4 B.C.
It thus appears without the shadow of a doubt that the day assigned the the Birth of the Lord, viz. December 25, was the day on which He was "begotten of the Holy Spirit", i.e. by pneuma hagion = divine power (Matt. 1:18, 20 marg.), and His birth took place on the 15th of Ethanim, September 29, in the year following, thus making beautifully clear the meaning of John 1:14, "The Word became flesh" (Matt. 1:18, 20) on the 1st Tebeth or December 25 (5 B.C.), "and tabernacled (Gr. eskeno-sen) with us", on 15th of Ethanim or September 29 (4 B.C.).
To sum up:
Jesus began His ministry during the time John the Baptist ministered in the wilderness, and John’s ministry started “in the fifteenth year of the reign of Tiberius Caesar, Pontius Pilate being governor of Judea, and Herod being tetrarch of Galilee, and his brother Philip tetrarch of the region of Ituraea and Trachonitis, and Lysanias tetrarch of Abilene, during the high priesthood of Annas and Caiaphas” (Luke 3:1-2). The only time period that fits all of these facts is A.D. 27-29. If Jesus was “about thirty years of age” by A.D. 27, a birth sometime between 6 and 4 B.C. would fit the chronology. More specifically, Jesus would have been approximately 32 years old at the time He began His ministry (still “about thirty years of age”). https://www.gotquestions.org/what-year-was-Jesus-born.html
Another useful source of information is this article by John Mosley on errors regarding the year in which Jesus was born: https://www.ips-planetarium.org/page/a_mosley1981
Section 3 discusses the Old Roman calendar which was based on the founding of Rome (AUC) but fell into disuse. Section 6 explains how the census was not for taxation purposes but to give an oath of allegiance, and this may have happened in 3 B.C.