What month was the King Herod census taken?

Because of the account in Luke's Gospel, historians have spent considerable effort in establishing the facts of this census, said to have occurred during the reign of King Herod and when Quirinius (or Cyrenius) was governor of Syria.

They find that, in spite of Luke 2:1, there never was a census during the reign of Augustus. They have also established that Quirinius only became governor of Syria in 6 CE, ten years after the death of King Herod, and that he appears never to have held a less prominent position in Syria previously. They find that the Jewish historian, Josephus says that the census under Quirinius was indeed the first Roman census taken in Palestine.

Scholars say that if there had been a Roman census during the time that Herod was alive, it would not have included Galilee or Judea, since Herod ruled autonomously and was responsible for raising his own taxes. In fact, the census described in the Gospel of Luke is a total anomaly.

Raymond E. Brown (An Introduction to the New Testament) says the best explanation is that, although Luke likes to set his Christian drama in the context of well-known events from antiquity, sometimes he does so inaccurately. There was no census in the time of King Herod. Naturally we can not say what month a non-existent census took place.