Early Quakers did not observe Christmas nor mark other 'times and seasons'. They believed that no day was more holy than any other, and believed that each day, and all of life, was sacred (Quaker Faith and Practice 27.39 and 27.42), which is not the same as saying they taught Christ was born in Autumn!
However, books have been written by some who believe Jesus was born in September. This was proposed by Ernest L. Martin in his book, The Star that Astonished the World (ASK Publications, 1996). He claims to have worked out the birth of Jesus to be precisely at 7pm on the 11th September 3 B.C.E. Not that I have read his book. Nor do I know if he is speaking for himself or as representative of any Christian denomination.
He is not he the only one either. I've read various papers also arguing for a late September or early October date. Unfortunately, I only kept one, but that was not written by anyone claiming to speak for a Christian denomination. The author was David Hughes, Professor of Astronomy at the University of Sheffield, England. An article appeared in the 1st September 2009 "Weekend" magazine, which was reviewing a BBC2 documentary due to be screened on Christmas Eve that year. Hughes identified the year for the triple conjunction of stars (which seemed to coincide with a little-known census of Quirinius) as several years earlier than 3 B.C.E. The first conjunction was in May, getting the Magi off on their long journey. The second conjunction would fit in with the star leading the Magi to Jerusalem in September, but the third conjunction happened a few weeks later in November - the one that led them to Bethlehem and the house that Joseph, Mary and 'the child' were then living in. This fits in with the view that many Christians have that the Magi never pitched up on the night of Jesus' birth, but some time later, meaning Jesus was born before that November conjunction - September or October being the contending months.
Most Christian denominations are willing to admit that the actual date of Jesus' birth is not stated in the Bible, and they allow leeway for individuals to hold to various opinions on the matter. After all, nobody's salvation depends on stating belief that Jesus was born in December! Just that he was conceived, miraculously, by the virgin, and born to the acclaim of angels, and the wonder of shepherds. It is the acclaim and wonder Christians still hold for that gift of God from heaven that is critically important - irrespective of what month of the year it happened.
One denomination that ties early October in with their condemnation of the date 25 December as the wrong date for his birth is the Jehovah's Witnesses. They have clearly stated their view that it was early October Jesus was born, as these two quotes show:
"The evidence, then is that Jesus' baptism and anointing took place in
the fall of the year 29 C.E. Since Jesus was "about thirty years old"
at the time of his baptism in 29 C.E. (Lu 3:23), his birth took place
30 years earlier, or about the fall of the year 2 B.C.E." (Insight on
the Scriptures Vol. 1, p467)
"Evidently, Jesus was born sometime in October." (Jesus, the Way, the
Truth, and the Life p18)
Other answers here show that the Restored Church of God and the United Church of God also have early autumn / the Feast of Tabernacles as the season of Jesus' birth.