There are in depth understandings for those who wish to dig...The Greek word for “first” in Luke 2:2 is 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.”
Another explanation could be, 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 his first term as governor, and another was ordered during his second term as governor mentioned in Acts 5:37, which probably took place between AD 6-7 (Josephus links this census to an uprising under Judas of Galilee). With Luke being the author of both Luke and Acts and wanting to write in “consecutive order” (Luke 1:3), it would seem unlikely for Luke to make such a mistake in dating.
There is a long list of famous historians, including Roman ones, that risked their reputations and in some cases, their very lives, on reporting the crucifixion and resurrection of Jesus Christ:
Pliny the Younger (62-114AD)
Gaius Suetonius Tranquillus (75-130AD)
Cornelius Tacitus (55-120AD)
Pontius Pilate’s wife’s written letter Recollections of the last occurrences from the Life of Jesus Christ.
Publius Lentulus (ruler of Judea) wrote a letter to Roman emperor Tiberius (resides in a private library in England, found 1865 AD).
Pontius Pilate, Roman governor, wrote a report to Roman Caesar Tiberius in Jerusalem.
Celsus (170 AD)
Lucian of Samoset (160 AD)
Tacitus (70 AD)
Others included Africanus, Origen, Suetonius, Pliny the Younger, and Eusebius of Caesarea, and even the Syrian philosopher, Mara Bar-Serapion.
It really boils down to this, a quote by Basal Pascal, who sums it up beautifully..."In faith there is enough light for those who want to believe and enough shadows to blind those who don't." For believers the question is, "why do you want to believe?" for non believers, it is, "why do you not want to believe?" There are answers to those questions, gotta dig deep and soul search!