Except there was no "Massacre of the Innocents." The nearest is either Herod the Great killing his own sons or Herod Archelaus killing 3,000 in the Temple.
From Wiki: [wiki]Quirinius[/wiki]
After the banishment of the ethnarch Herod Archelaus in 6 AD, Iudaea (the conglomeration of Samaria, Judea and Idumea) came under direct Roman administration with Coponius as prefect; at the same time Quirinius was appointed Legate of Syria, with instructions to assess Iudea Province for taxation purposes. One of his first duties was to carry out a census as part of this.
We see that the census was after the reigns of both Herod the Great and Herod Archelaus (the former died 4BC and the latter was removed in 6AD.)
So that puts the census after any rule of any Herod.
The Gospel of Luke links the birth of Jesus to a "world-wide" census ordered by Augustus carried out while Quirinius was governor of Syria. This is thought to be a reference to the census of Judea in 6/7 AD; however, Luke also, like the Gospel of Matthew, dates the birth to the reign of Herod the Great, who died in 4 BC, ten years before the census of 6 or 7 AD. According to Raymond E. Brown, most modern historians suggest that Luke's account is mistaken. Scholars trying to reconcile the Biblical accounts have speculated about alternative explanations, such as a different census before Quirinius was governor. The majority view, however, among modern scholars is that there was only one census, in 6, and the author of the Gospel of Luke deviated from history in connecting it with the birth of Jesus.