Not at all. Luke used the term "hegemon", which is a broader term than "governor", and may well be referring to the administrative role Cyrenius (Quirinius) was assigned as opposed to being titled as "governor". Cyrenius' history is indeed well documented. He ruled both Syria and Judea after the year 6 A. D. when Archelaus was deposed as king of Judea. Both scripture and Josephus indicate this was well after Jesus’ birth and Herod’s death (Mat 2:1,16,22; Antiquities 17.8.1-4, 17.13.1-5;18.1.1-6). Armenian historian Moses of Khorene (Armenian History 2:26) says that in 3 B. C. Roman authorities came to Armenia to set up images of Caesar Augustus in the temples of the area. These same sources state that it was the registration mentioned in Luke which brought them there. The purpose of this registration was to record an official declaration of allegiance from all of his subjects to present to Caesar Augustus in celebration of his Silver Jubilee. History records that Varus was governor of Syria from about 7 B.C. to about 4 B.C. and was not a trustworthy leader. However, Cyrenius was a notable military leader. During the census of 8-7 B.C., Augustus entrusted Cyrenius with Palestine, effectively superseding the authority and governorship of Varus by appointing Cyrenius to a place of special authority. Cyrenius administered in Syria on two separate occasions, once while prosecuting the military action against the Homonadensians between 12 and 2 B.C., and later beginning about A.D. 6. A Latin inscription discovered in 1764 has been interpreted to refer to Cyrenius as having served as governor of Syria on two occasions. History records that Cyrenius was on assignment in Syria during this time and was one of the few trusted leaders. It is probable that Varus was on his way out while Cyrenius was taking charge of matters during Luke's narration. Luke has proven himself over and over throughout the centuries to be a reliable historian, even in the details. In his making reference to 32 countries, 54 cities, and 9 islands, Luke made no mistakes.

When Pilate saw that he could prevail nothing, but that rather a tumult was made, he took water, and washed his hands before the multitude, saying, I am innocent of the blood of this just person: see ye to it. Then answered all the people, and said, His blood be on us, and on our children. Then released he Bar-ab'-bas unto them: and when he had scourged Jesus, he delivered him to be crucified. Then the soldiers of the governor took Jesus into the common hall, and gathered unto him the whole band of soldiers. And they stripped him, and put on him a scarlet robe. And when they had platted a crown of thorns, they put it upon his head, and a reed in his right hand: and they bowed the knee before him, and mocked him, saying, Hail, King of the Jews! And they spit upon him, and took the reed, and smote him on the head. And after that they had mocked him, they took the robe off from him, and put his own raiment on him, and led him away to crucify him. Matthew 27:24-31

An enrolment was due in Syria in the year 8 B.C. and made; but in Herod's kingdom it was probably delayed for some time, for Herod had gotten himself into trouble with Augustus. With. the consent of Saturninus, governor of Syria, Herod had marched an army into Arabia to redress certain wrongs which he had received (Ant., XVI., ix. 2). This proceeding was misrepresented to the emperor, who notified Herod, probably in the year 8 B.C., that henceforth he would treat him as a subject. Some time afterward the whole nation of the Jews, except 6,000 Pharisees, took an oath of fidelity to Caesar and the king jointly (Ant., XVII, ii. 4). Obviously the two acts, the oath and the enrolment, form part of the new policy of Augustus toward Herod. The date of the enrolment and the oath may be the year 6 B.C.; for Herod would have had little difficulty in obtaining leave from Saturninus to postpone the numbering until the embassy, which, after Augustus announced the change of policy toward him, he was sending to Rome to seek a reconciliation with the emperor and a restoration of the old order, should return and report the result of its efforts. Herod was finally obliged to order the census, and it was probably taken in the summer of the year 6 B.C., when Quirinius was a special legatos Aogusti to Syria, invested with the command of the army and entrusted with its foreign affairs, such as the relations between its several states and Rome, particularly where tension existed and military intervention might be necessary. Quirinius stood in exactly the same relation to Varus, the governor of Syria, as at a later time Vespasian did to Mucianus. Vespasian conducted the war in Palestine while Mucianus was governor of Syria; and Vespasian was Legatus Augusti, holding precisely the same title and technical rank as Mucianus

in addition to what's been said, we know that the year we have assigned to be Jesus's birth year is off by about 4-6 years.

No, Luke did not make an error. Hopefully these solutions will help. First, Quintilius Varus was governor of Syria from about 7 B.C to about 4 B.C. Varus was not avery trustworthy leader, a fact that was disastrously demonstrated in A.D 9 when he lost three legions of soldiers in the Teutoburger forest in Germany. To the contrary, Quirinias was a notable military leader who was responsible for stopping the rebellion of the Homonadensians in Asia Minor. When it came time to begin the census, in about 8 or 7 B.C., Augustas entrusted Quirinius with the delicate problem in the volatile area of Palestine, effectively superseding the authority and governorship of Varnus by appointing Quirinius to a place of special authority in this matter. It has also been proposed that Quirinius was governor of Syria on two seperate occasions, once while prosecuting the millitary action against the Homonadensians between 12 and 2 B.C, and later beginning about A.D. 6. A Latin inscription discovered in 1764 has been interpreted to refer to Quirinius as having served as governor of Syria on two occasions. It is possible that Luke 2:2 reads, "This census took place before Quirinius was governing Syria." In this case, the Greek word translated "first" (protos) is translated as a comparative, "before." Because of the awkward construction of the sentence, this is not an unlikely reading. Regardless of what solution is accepted, it is not necessary to conclude that Luke has made an error in recording the historical events surrounding the birth of Jesus. Luke has proven himself to be a reliable historian even in the details. It has been shown that in making references to 32 countries, 54 cities, and 9 islands he made no mistakes. I go with Luke.