Yes, the Eastern Churches in the Byzantine tradition all start the new liturgical year on September 1.
Here is the Gospel reading heard on Sunday:
16 And he came to Nazareth, where he had been brought up; and he went to the synagogue, as his custom was, on the sabbath day. And he stood up to read; 17 and there was given to him the book of the prophet Isaiah. He opened the book and found the place where it was written,
18 “The Spirit of the Lord is upon me,
because he has anointed me to preach good news to the poor.
He has sent me to proclaim release to the captives
and recovering of sight to the blind,
to set at liberty those who are oppressed,
19 to proclaim the acceptable year of the Lord.”20 And he closed the book, and gave it back to the attendant, and sat down; and the eyes of all in the synagogue were fixed on him. 21 And he began to say to them, “Today this scripture has been fulfilled in your hearing.” 22 And all spoke well of him, and wondered at the gracious words which proceeded out of his mouth; and they said, “Is not this Joseph’s son?”
The Melkite commentary on the Menaion first notes the date itself has a connection to the old empire:
Starting in the last half of the fifth century (probably AD 462), the Byzantine Empire designated September 1 as the first day of the New Year. The Byzantine liturgical year was arranged according to that calendar and September 1 remains the first day of our liturgical year. The cycle of the Church’s Great Feasts begin in September with the Nativity of the Theotokos (September 8) and conclude in August with the feast of her Dormition (August 15).
There are two more terms that have been traditionally associated with the turning of the calendar: the “Indiction,” and the “Year of the World.”
Again, from the Menaion commentary, which makes clear why the particular Gospel reading above starts off the new year:
The Indiction
The first day of the Church year is called the Indiction. Originally referring to the start of a tax assessment cycle in the Roman Empire, this word has come to mean the beginning of a cycle in a more general way and may be found in legal or formal documents to this day. Thus in 2011 Pope Benedict XVI issued a formal letter “For the Indiction [i.e. Beginning] of the Year of Faith.” And so calling September 1 an Indiction simply means that it is the start of a new cycle of the feasts, fasts and other observances of our Church.
On this day Byzantine churches read the Gospel of the beginning of Christ’s public ministry as recorded in Luke 4:16-22. After reading the Messianic prophecy in Isaiah 61:1-2 the Lord tells His listeners, “Today this Scripture is fulfilled in your hearing.” The Messiah is at hand: God’s plan is on the move.
The “Year of the World”
A lesser-known aspect of the Byzantine calendar is that September 1, AD 2024 (1) is the first day of AM 7533. From AD 691 to 1728 the Byzantine Churches followed a system dating years from the creation of the world according to the calculations in the Book of Genesis (AM, Anno Mundi, the “Year of the World”).
In 1700, during his westernization of Russia, Tsar Peter the Great replaced the Byzantine Era in his realm with the Western Christian Era. A few years later the Patriarchate of Constantinople and all the Churches in the Ottoman Empire followed suit.
And coming up soon, two of the Twelve Great Feasts in rapid succession: the Nativity of the Theotokos and the Exaltation of the Cross. Hitting the ground running, it seems.
Footnote:
(1) The dates have been updated from the original publication in 2013.