Answer: Unlikely. Shepherds were out tending flocks in the field? This would be the rainy season in Israel, so check out what the Bible says in Luke 2:1-7.
Luke 2:1-7," And it came to pass in those days that a decree went out from Caesar Augustus that all the world should be registered. This census first took place while Quirinius was governing Syria. So all went to be registered, everyone to his own city. Joseph also went up from Galilee, out of the city of Nazareth, into Judea, to the city of David, which is called Bethlehem, because he was of the house and lineage of David, to be registered with Mary, his betrothed wife, who was with child. So it was, that while they were there, the days were completed for her to be delivered. And she brought forth her firstborn Son, and wrapped Him in swaddling cloths, and laid Him in a manger, because there was no room for them in the inn."
If I were a Roman ruler looking for tax revenue, I would try to make the census successful by picking a time that would get the money to me as quickly and easily as possible. Summoning people in the middle of the rainy season, or the cold Winter, would be impractical—and unpopular. Maybe the best time would be to summon them after the Fall harvest, after Summer has passed. Crops are in the barns, and the weather is still good.
Interestingly, this timing would match with what we know from the account of John the Baptist's father. John the Baptist was born six months earlier than Jesus. And his father Zacharias performed service in the temple at appointed times Luke 1:35-37, "And the angel answered and said to her, 'The Holy Spirit will come upon you, and the power of the Highest will overshadow you; therefore, also, that Holy One who is to be born will be called the Son of God. Now indeed, Elizabeth your relative has also conceived a son in her old age; and this is now the sixth month for her who was called barren.'"
When you look into the Bible's account of the Temple service of Zacharias, John's father, you see that the timing corresponds to a Spring birthdate for John—and thus a Fall birth, not a Winter birth, for Jesus.