Could Jesus have been born in September instead of December?

We do not know what time of year he was born. The reason for December is due to the Christianization of the Roman world under the Emperor Constantine. Therefore, to draw a dogmatic conclusion on the season of his birth is only speculative. Here is my best guess- early spring. Why do I believe this? Because of where shepherds would have been grazing their sheep. It is green near Bethlehem and Jerusalem in early spring. Albeit, this year, they are getting a lot of rain in December! So, my best guess is based on grazing patterns of sheep.

That being said, I don't go to Christmas parties and poo-poo the dating. I just enjoy my Egg Nog and celebrate Jesus. We don't know the date for sure, so may as well just enjoy what we already have.

I agree with user15740, the reason for the 25th December was due to the Christianization of the Roman Empire, more specifically to include one of their Gods, Mithras, who was born on the 25th December and was celebrated by giving gifts to one another on the 25 December and was worshiped on Sundays. https://www.livescience.com/61309-roman-temple-sun-alignment-christ-birth.html

And also December is lambing season in Isreal and the middle of winter. Shephards would not take sheep out in the fields, it is cold and they would have their hands full with sheep giving birth anywhere. I grew up part-time on a farm and during lambing season the livestock was kept in the barn or where they ever they usually would spend nighttime. https://www.fao.org/3/p8550e/P8550E01.htm

The Fall makes far more sense than late December. As I've written elsewhere:

Elisabeth conceived (John the Baptist) immediately after Zacharias finished his Abia priestly duties (Luke 1):

[5] ... priest named Zacharias, of the course of Abia ... [8] ... he executed the priest's office before God in the order of his course [23] ... as soon as the days of his ministration were accomplished, he departed to his own house. [24] And after those days his wife Elisabeth conceived ...

Her cousin Mary conceived (Jesus) six months later:

[26] And in the sixth month the angel Gabriel was sent from God unto a city of Galilee, named Nazareth [27] To a virgin espoused to a man whose name was Joseph, of the house of David; and the virgin's name was Mary. [36] And, behold, thy cousin Elisabeth, she hath also conceived a son in her old age: and this is the sixth month with her, who was called barren. [40] And [Mary] entered into the house of Zacharias, and saluted Elisabeth. [56] And Mary abode with her about three months ...

The course of Abia is defined in the First book of Chronicles:

[24:10] The seventh to Hakkoz, the eighth to Abijah [24:19] These were the orderings of them in their service ...

Each priest served for a week, with the cycle beginning on the 1st day of Nisan. But during the week of Passover (Nisan 14) and Pentecost (7 weeks later) all priests served together, so Zacharias, being of the 8th course, would have finished his first course after the 10th week of the year. Six months after that (Jesus's conception) would have been around what the Romans called late December, and nine months after that (Jesus's birth) would have been late September or early October.

That would be around the time of Sukkot (the Feast of Tabernacles). Sukkot is one of three times each year when people made a pilgrimage to Jerusalem. In addition to the normal tithe given to support the priesthood, people were encouraged to save a second tithe to use for their own expenses and celebrations during these times (Deuteronomy 14:26):

And thou shalt bestow that money for whatsoever thy soul lusteth after, for oxen, or for sheep, or for wine, or for strong drink, or for whatsoever thy soul desireth: and thou shalt eat there before the LORD thy God, and thou shalt rejoice, thou, and thine household.

Even today, most people that observe this fall festival travel away from home, or if they can't, they arrange to sleep in a tent or some other location different from their normal home.

It's by no means a certainty that this is when Jesus was born, but if Jesus had been born during the Feast of Tabernacles, it then confirms why his parents stayed in their family's home town of Bethlehem, just south of Jerusalem.

There's a couple of issues when trying to fit the 24 priestly courses into a temple calendar. One the Jewish calendar is lunisolar. Two, which is related, is the necessity about each three years to add a month (intercalary) so that the moon phase, which is slower, catches up to the solar phase. So, in the OP it appears the assumption is made that the priestly course starts the same each year. Zechariah was assumed to be there in the 9th week or about June (Sivan). John would have been born nine months later in March (Nisan) and Jesus six months later in September.

But he also would have been there in their month Kislev or our December. Nine months later John would be born in September and Jesus would have been born 6 months later about Passover in Nisan. This idea of being born and dying on the same day apparently grew in the Church in the 3rd century (source).

And again, add in the rotation of the priestly courses where 48 plus 2 (Passover, Tabernacles) and the 50 weeks plus adding the three-year intercalary month and it is a huge assumption on when Zechariah was serving. I would add, however, that the 69 weeks of the 70 weeks of Daniel using the prophetic year (see Sir Robert Anderson) provides some support for the OP's assumption.

So, to answer the OP, the easiest reply to the priestly course "proof" is to show the huge assumption that prior to Jesus' birth in that particular year, Zechariah would have been present in the 9th week of the year, while ignoring he could also have been present in early December, even though this fact does not support a 12/25 birth, but rather a Nisan birth. And ignoring the fact that about each three years, the timing of the priestly course changes. Zechariah could just as easily been present in September and John born in June and Jesus born 6 months later in December.

Yes, Jesus could have been born in the Fall as well as in late December. The evidence is inconclusive. The article Was Jesus Really Born on December 25th by John J. Parsons from the Hebrew for Christians website has arguments for both Fall and late December birth.

Argument for the late December birth:

1. Assume that Zechariah was told by the angel during Yom Kippur (Tishri 10) and John the Baptist was conceived within days afterwards. He would have been born 40 weeks later in Tammuz (June/July).

2. Jesus was conceived 6 months after John the Baptist was conceived (Luke 1:24-27, 36), thus in Nisan (Mar/Apr) near Passover.

3. 40 weeks later, Jesus would have been born in late December (Tevet).

4. Church history as early as the late first century has attested a late December birth, then Hippolytus (2nd century), then John Chrysostom (347-407) as well. Quote from St. Hippolytus's Commentary on Daniel, c. 204 AD:

   For the first advent of our Lord in the flesh, when he was born in Bethlehem, was December 25th, a Wednesday, while Augustus was in his forty-second year, but from Adam, five thousand and five hundred years.
   _{Source: The 25th of December: Pagan Feast or Patristic Tradition?}

5. Early Jewish sources suggest that the sheep around Bethlehem were outside year-round.

For more details please visit the article linked above.

Another article When is Jesus' Birthday? by Brent MacDonald discussed these 2 scenarios in greater detail as possibility #2 (Fall birth) and possibility #3 (December birth) respectively. In addition he describes possibility #1, #4-#6 which he rejected. Like John Parsons, he also leaning towards the Fall birth scenario.