Would “This enrolling first took place managed by Quirinius of Syria” be more accurate than “when Quirinius was governor”?

No.

The participle ἡγεμονεύοντος and its subject Κυρηνίου are in the genitive. That makes this a Genitive absolute, which corresponds to the Latin ablative absolute. Its job is to indicate the time or circumstance under which the main clause happened: it is a separate clause. English equivalents are Absolute constructions, such as The referee having finally arrived, the game began; All things considered, it’s not a bad idea. But in Latin and Greek, grammatical case is used to differentiate the two clauses (as well as the participle in the dependent clause).

So the main clause is “this enrolling first took place”. But the following participle means “under the circumstance of Quirinius managing Syria”. Greek wouldn’t use “of Syria” to describe a person, but “Syrian” or “from Syria”: the genitive for Syria is in fact the object of hegemoneuō “ruling, managing”. And it’s a separate clause, so the enrolling has nothing to do with the managing.

So: “With Quirinius managing Syria, this enrolling first took place.”

Which pretty much is the same thing as “When Quirinius was governor of Syria.”