Yes, Quirinius was a governor around that time, while Herod was still alive, but as a legate in Galatia, not in Syria.
He was a general of the army that operated in both Galatia and Cilicia, and fought the insurrection of Homonadenses, from 12 - 1 BCE. He was too busy to also govern Syria and hold a census That’s what creationists tends to ignore.
During the last 6 years of Herod’s life, two different governors served in Syria at that time:
- Gaius Sentius Saturninus (9 - 7/6 BCE)
- Publius Quinctilius Varus (7/6 - 4 BCE)
Josephus does mention Saturninus and then Varus succeeding Saturninus (AotJ Book 17, chapters 1, 2 & 5). Varus is found in the works of Publius Cornelius Tacitus and of Marcus Velleius Paterculus (Roman History, book II, chapter 117)
But Varus was a son-in-law of Marcus Vipsanius Agrippa (Augustus closest friend), and most famous being general in Germania in 9 CE, where he lost 3 of his legions and his own life in ambush at the Teutoberg Forest.
A Christian historian named Tertullian, on the other hand, say Jesus was born when Saturninus was governor (Against Marcion, 4, 19).
If Jesus was indeed born when Herod was alive, then it would occur when Varus was governor of Syria (7/6- 4 BCE), not Quirinius (AotJ Book 18, chapter 1, Quirinius’ governorship and census held 10 years after Herod’s death and after Archelaus was banished in 6 CE).