Exegesis:

hautē apographē prōtē egeneto hēgemoneuontos tēs Surias Kurēniou ‘this registration, the first, happened when Quirinius governed Syria’; the phrase serves to indicate (1) the time when the registration referred to in v. 1, took place, and (2) the fact that this registration was the first to happen. Hence a rendering like “this was the first registration of its kind; it took place when Quirinius was governor of Syria” (New English Bible, cf. Zürcher Bibel) seems to cover best the full implications of the Greek text. Registrations in the Roman empire might easily extend over a period of many years and Quirinius acted as the special representative of the Roman emperor and was engaged in a census from 12 B.C. to 16 A.D.; during this period he was also twice governor of Syria, probably in 3-2 B.C. and certainly from 6 A.D. to the end of his term as special representative; for a full discussion cf. IDB III, 975ff.

apographē ‘registration,’ ‘census,’ here according to Stauffer (op. cit. p. 29) referring to “the systematic registration of all taxable persons and of all things subject to taxes”; in most cases the latter had to do with landed property, and since Joseph was of the house and lineage of David he very probably would have a share in the landed property of the descendants of David in his place of origin (op. cit. p. 33).

hēgemoneuontos tēs Surias Kurēniou ‘when Quirinius governed Syria’; the phrase may either mean that the registration of Palestine took place during Quirinius’ first period of actual governorship of the Roman province of Syria or contain a somewhat incorrect reference to the whole period of his activities in the East.

hēgemoneuō (also 3.1) ‘to govern,’ ‘to rule,’ a very general word, alike applicable to the Roman emperor and to his underlings in the provinces of the empire.

Kurēnios Greek transcription of the Roman name ‘Quirinius.’

Translation:

This was the first enrollment, or, shifting to a verbal construction, e.g. ‘this was the first time people had to enroll themselves’; or again, using a generic reference to the verb in v. 1, ‘this happened (or, was done) for the first time.’ Enrollment, cf. on the verb in v. 1; most renderings refer only to the persons registered, some include their possessions, e.g. ‘numbering-of-houses (referring to population and live-stock)’ (Kannada).

When, preferably, ‘done/held when,’ ‘it happened when.’

Quirinius. Older versions transliterate the Greek form of the name, but it is preferable to take the original Latin form as basis of transliteration; usually qu then can best become k, or its closest equivalent.

Governor. Usually one can find an appropriate equivalent in the local or national official hierarchy of the present or in a colonial one of the past, e.g. ‘province-ruler’ (Tamil, Telugu, Kannada), ‘the exalted person’ (Toraja-Sa’dan, using the title of the Governor-General and of provincial governors); or, since Quirinius represented the imperial government, ‘representative of the government’ (Bahasa Indonesia), cf. ‘stood in power … representing the Emperor’ (Balinese).

Quoted with permission from Reiling, J. and Swellengrebel, J.L. A Handbook on the Gospel of Luke. (UBS Handbook Series). New York: UBS, 1971. For this and other handbooks for translators see here . Make sure to also consult the Handbook on the Gospel of Mark for parallel or similar verses.