In regards to the accusation of Luke being contradictory in the timeline of Christ’s birth, this has been well studied and rigorously explained by many others.
One such study is provided below from http://www.christian-thinktank.com/quirinius.html
There are two main interpretations of this MS: one is that it refers to Q. Varus (placing Quirinius as a procurator during the birth of Christ), and the other that it refers to Quirinius himself.
The first option is defended by Ernest Martin in CKC:90:
” A Latin inscription found in 1764 about one-half mile south of the ancient villa of Quintilius Varus (at Tivoli, 20 miles east of Rome) states that the subject of the inscription had twice been governor of Syria. This can only refer to Quintilius Varus, who was Syrian governor at two different times. Numismatic evidence shows he ruled Syria from 6 to 4 B.C., and other historical evidence indicates that Varus was again governor from 2 B.C. to A.D. I. Between his two governorships was Sentius Saturninus, whose tenure lasted from 4 to 2 B.C. Significantly, Tertullian (third century) said the imperial records showed that censuses were conducted in Judea during the time of Sentius Saturninus. (Against Marcion 4:7). Tertullian also placed the birth of Jesus in 3 or 2 B.C. This is precisely when Saturninus would have been governor according to my new interpretation. That the Gospel of Luke says Quirinius was governor of Syria when the census was taken is resolved by Justin Martyr’s statement (second century) that Quirinius was only a procurator (not governor) of the province (Apology 1:34). In other words, he was simply an assistant to Saturninus, who was the actual governor as Tertullian stated.”
The second option is favored by William Ramsey (NBD, s.v. “Quirinius”):
“The possibility that Quirinius may have been governor of Syria on an earlier occasion (*Chronology of the NT) has found confirmation in the eyes of a number of scholars (especially W. M. Ramsay) from the testimony of the Lapis Tiburtinus (CIL, 14. 3613). This inscription, recording the career of a distinguished Roman officer, is unfortunately mutilated, so that the officer’s name is missing, but from the details that survive he could very well be Quirinius. It contains a statement that when he became imperial legate of Syria he entered upon that office ‘for the second time’ (Lat. iterum). The question is: did he become imperial legate of Syria for the second time, or did he simply receive an imperial legateship for the second time, having governed another province in that capacity on the earlier occasion?...The wording is ambiguous. Ramsay held that he was appointed an additional legate of Syria between 10 and 7 bc, for the purpose of conducting the Homanadensian war, while the civil administration of the province was in the hands of other governors, including Sentius Saturninus (8-6 bc), under whom, according to Tertullian (Adv. Marc. 4. 19), the census of Lk. 2:1ff. was held.
Under either of these scenarios, SOMEONE served twice, and under either of these scenarios, Quirinius could EASILY have been responsible for the census.
And curiously enough, even if that were NOT the case somehow, the linguistic data of the last few decades indicates that Luke 2.1 should be translated ‘BEFORE the census of Quirinius’ instead of the customary ‘FIRST census of Quirinius’—see Nigel Turner, Grammatical Insights into the New Testament, T&T Clark: 1966, pp. 23,24 and Syntax, p. 32. This would ‘solve the problem’ without even requiring two terms of office for Q.
And, while we are talking about Greek here...the term Luke uses for Quirinius’ ‘governorship’ is the VERY general term hegemon, which in extra-biblical Greek was applied to prefects, provincial governors, and even Caesar himself. In the NT it is similarly used as a ‘wide’ term, applying to procurators—pilate, festus, felix—and to general ‘rulers’ (Mt 2.6). [The New Intl. Dict. of New Test. Theology (ed. Brown) gives as the range of meaning: “leader, commander, chief” (vol 1.270)...this term would have applied to Quirinius at MANY times in his political career, and as a general term, Syria would have had several individuals that could be properly so addressed at the same time. Remember, Justin Martyr called him ‘procurator’ in Apology 1:34, which is also covered by this term.] My point is...nothing is really out of order here...
First, Luke 2:1 says KyrhnioV (pron. Kyrenios), which means Quirenius in Latin. No confusion there.
Second, the NT tells us that Jesus was born during the life of Herod the Great. This Herod died in 4 BC.
Third, another Herod (Arhelaus), succeeded Herod the Great (Archelaus' father) and ruled as an etnarch of Samaria, Judea and Edom between 4 BC and 6 AD, when he was exiled to Gaul and Judea was turned into a Roman province, and Quinirius was tasked with compiling a census of the region.
Luke clearly states that it was under Quinirius (not Quintilius!) "everyone was traveling" (v. 2:3) in order to register for the census ordered by Quintilius. It then says (verse 2:4) literally "Now went up also Joseph...and Mary, the one having been engaged to him, was pregnant [εγκυω]," in order to register.
There can be no doubt when this happened. There was no census in that aprt of the region untl 6 AD.
Yet Matthew 2:13-23 speaks of Jesus being alive when Herod (the Great) was still living (before 4 BC). The problem is that Luke contradicts himself (1:5) and is agreeing with Matthew!
Fourth, according to Matthew, Herod ordered the slaughter of the Innocents. Tere is no record of any kind that this happened, not even by Jospehus who wa sin a habit of describing thing about Herod in great detail.
Under either of these scenarios, SOMEONE served twice, and under either of these scenarios, Quirinius could EASILY have been responsible for the census There would have been no need for census until Judea was made a Roman province, and this happened in 6 AD. Matthew's accounts says Jospeh and Mary and her Son went to Egypt but were told to go to Nazareth after Herod died.
Accoridng to Luke (2:1), however, they went to Nazareth to be registered because everyone went to their respective cities. No angel calling them out of Egypt!
Accrding to Luke 1:5 a completely different sotry develops. We have a clear inter-author and inter-bible contradiction or, to use a PC term, disagreement.