How could Jesus be born during the reign of Herod, and when Quirinius was governor of Syria, if those periods were not contemporary? In Matthew 2, it makes it clear that Jesus was conceived during the reign of Herod and was a young child when Herod died.


  After Herod died, an angel of the Lord appeared in a dream to Joseph in Egypt and said, “Get up, take the child and his mother and go to the land of Israel, for those who were trying to take the child’s life are dead.”


We know that Herod died in 4 BC, so we know that Jesus must have been born in 4 BC or shortly beforehand.

In Luke 2, it is made clear that Jesus was born during the census of Quirinius


  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.) And everyone went to their own town to register. ...
  He went there to register with Mary, who was pledged to be married to him and was expecting a child. While they were there, the time came for the baby to be born, and she gave birth to her firstborn, a son.


Quirinius became governor of Syria and performed his census in 6 AD. From this we know that Jesus must have been born in 6 AD.

This is a nine year discrepancy between the time of Jesus' birth as recorded by Matthew and the time of Jesus' birth as recorded by Luke. How can these two accounts be reconciled?
This is a fairly common question, and there is a very good answer.  A detailed answer can be found here and here.  These are some of the highlights.

It is important to note that Luke mentions that the census to which he is referring is the first census taken while Quirinius was governing.  This seems to indicate that at the time of writing, the readers would need to distinguish between multiple censuses.  Luke specifies that this was the first one.

Additionally, the term used for the ruling of Quirinius in Luke is not specific to a certain position.  So, it's quite possible that he held one position at first and then was promoted to the specific position of governor and performed a census then as well.

So, linking the census mentioned by Josephus to this first census that took place when Quirinius was in a position of governing is uncertain at best.

Being 2,000 years removed from this, our historical data and understanding are certainly not what they would have been to first century readers of Luke's account.  It is doubtful that Luke would have created such a glaring inconsistency, and if there were one, the readers of that day would surely have raised objection to it and discounted the narrative.  Indeed, there were many in that day who would have loved to expose Christianity as false, if given the chance.

Still, there is ample evidence to resolve this apparent discrepency when we pull back from the assumption that there was only one census that must be connected between Josephus and Luke, and we also understand that Luke does not specify a particular position in mentioning the governance of Quirinius.