The Greek translation of the Hebrew OT was called the Septuagint and I’ve read it was used in the early NT church and often quoted by NT writers. I’ve also read that it contained the Apocrypha, which was not accepted into the Hebrew OT canon. Question #1 “Was the Septuagint read in the NT synagogues?” Question #2 “If the Apocrypha wasn’t part of the Hebrew OT Canon, why would the Jews accept the Greek translation with it included?”
The Septuagint Greek translation was not an absolutely fixed set of documents for the Jews leading up to the time of the early church. At the time that the Hebrew Bible was translated into Greek, in the late third and early second century BC, a number of other Hebrew writings were also translated into the Greek. Some Jewish writings were even originally in Greek and did not have to be translated. This includes books such as Judith, Tobit, Wisdom, Ecclesiasticus and others. To make the situation a bit more confusing, the Jews did not have the identical concept of inspiration that we in the West are used to. To them, some Hebrew writings were super-inspired (Genesis-Deuteronomy), others were definitely inspired (Isaiah, Psalms, etc.) and others were somewhat inspired (1 Maccabees, 1 Enoch, etc.). They did not have the black and white idea of inspiration we are used to.
Jesus and the apostles appear not to have been confused about which of the Greek texts were canonical. Jesus quoted from the Septuagint (or he gave an Aramaic equivalent of the Septuagint), but he never quoted the Apocrypha. Neither did Paul of Peter, at least in their canonical writings. Jude probably quoted from 1 Enoch, but his is the only use of the Old Testament Apocrypha in the New Testament. The early church did not make such a hard and fast distinction. By as early as the second century, Church fathers were quoting from the Old Testament Apocrypha, using it in ways which imply they may have seen these writings as inspired. By the fourth and fifth centuries, the Apocrypha was used extensively and treated as essentially (although perhaps not quite) equivalent to the canonical Old Testament. This explains why Jerome included translations of the Greek Apocryphal writings in his Latin Vulgate translation. He did this under protest, only because he was forced to, because he was quite clear that these were not canonical. His was the minority view at the time. Because the Apocrypha was already widely accepted, and because it was included in the Latin Vulgate, the Old Testament Apocrypha was accepted as canonical by the Roman Catholic Church. It was also accepted by the Greek Orthodox and the Coptic Churches. Unknown to most Christians today, it was also accepted by the early Protestant leaders as well, by the way. It was only rejected outright by Protestants in the 1700s. Jerome believed that books like Judith and the additions to Daniel were not canonical in part, at least, because they were never in Hebrew. Later Protestants agreed with Jerome.
The Jewish attitude toward these books was quite different from that of the Christian Church. Before the time of Jesus, they did not consider 1 Enoch, 2 Maccabees and the like to be inspired in the same sense as Jeremiah or Kings, as evidenced by their writings, and by their use (or lack thereof) by Jesus. However, they might have used these documents as sub-inspired. At the Council of Jamnia (AD 90) they definitively rejected the Apocrypha and only included the books most of us recognize today as being the Old Testament. Over time, the Jews became even more adamant. Ironically, this was probably at least partially because of revulsion for the fact that the Christians were using the Apocryphal writings interchangeably with the canonical Old Testament. The lack of good attention to the Old Testament canon by the Christian Church had the reverse effect of crystallizing the Jewish/Hebrew canon.
Getting to your question: Was the Septuagint read in the New Testament synagogues? There were no “New Testament synagogues”, so this question is a bit confusing. Perhaps what you are asking is whether the Jews read from the Septuagint in the Jewish Synagogues during the time of Christ and immediately after that time. The answer is that then, as now, they only read from the Hebrew in the Synagogue. However, the Jews did use the Septuagint for studying the scriptures outside the Synagogue services. For the Jews whose first language was Greek (such as in Alexandria and people like Stephen in the New Testament), it was their principle Bible. Your second question is the harder one. The wording of your question implies that the Septuagint included the Apocrypha. Because the Septuagint is not precisely definable, this creates the ambiguity concerning the question you are asking. Let us not worry about the definition of the Septuagint. Let me put it this way. The Jews did use the Septuagint, but they did not accept as canonical those writings which we call the Old Testament Apocrypha. They may well have read 1 Enoch or Baruch in Greek, but when they did so, they did not read them as inspired in the sense that the canonical books were considered inspired. Bottom line, the Jews DID accept as inspired the Septuagint Greek translation of the Hebrew Bible, but DID NOT accept the Greek translation (or the original Greek, if it was not translated because it was never in Hebrew) of what we call the Apocrypha.
To use a rather clumsy analogy, if a person were to walk into your library at home and find there translations from the Greek of both Isaiah and 1 Enoch, they might mistakenly assume that any text you have translated from the Greek is considered by you to be part of the canon and inspired. Why else would you have translations of these books? You would, or course, reject this notion. However, perhaps your children might read both books and, over time, their descendants would no longer make a clear division between Isaiah and 1 Enoch. This would be analogous to what happened to the Christian Church (but not to the Jews!).
I hope this helps.