Can you cite some examples of these NT quotes and how they refer to the Greek version and not the Masoretic text?
Here’s one example: Matthew 12:18-21, quoting Isaiah 42:1-4. Where the Masoretic text has “He will not fail or be discouraged / till he has established justice in the earth; /
and the coastlands wait for his law” (RSV), Matthew (following the Greek reading) has: “and in his name the nations will hope.” Another example is Matthew 21:16: “Out of the mouth of babes and sucklings
thou hast brought perfect praise” as opposed to “Thou whose glory above the heavens is chanted / By the mouth of babes and infants, /
Thou hast founded a bulwark because of thy foes …” of the Hebrew. (The general rule is that when Matthew quotes from Isaiah - which he often does - he uses the Septuagint version.)
Luke also uses the Greek translation of Isaiah both when he applies Isaiah 40:3-5 to John the Baptist (Luke/LXX: “and all flesh shall see
the salvation of God” as opposed to the Hebrew’s “and all flesh shall see [it]
together”) and when he has Jesus read from Isaiah 61 in Luke 4:18-19. The Hebrew version of chapter 61 goes like this (RSV again):
The Spirit of the LORD God is upon me,
because the LORD has anointed me
to bring good tidings to the afflicted;
he has sent me to bind up the brokenhearted,
to proclaim liberty to the captives,
and the opening of the prison to those who are bound;
to proclaim the year of the LORD’s favor, …]
Luke’s version, however, goes:
“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,
to proclaim the acceptable year of the Lord.”
Here’s a nice list. And
another good one.
And since Jesus and the Apostles spoke, most likely, Aramaic, how is it that the author knew that they were citing the Greek version of the OT?
I hope you’re not imagining here that the NT authors jotted down every actual bit of dialogue Jesus and the apostles spoke word-for-word, as if they’re listening to a tape recording, translating what they heard word-for-word, and then jotting them down.

I think one of the reasons the authors used the Greek version is because it’s convenient: I mean, even nowadays, whenever we cite Scripture, most of us wouldn’t bother translating the original texts on our own, right? Very often we would just often cite from particular translations (or sometimes, if we are familiar with the passage, we go by memory - very often using the wording of one or more preexisting translations). Same with the NT authors: most of them used the Septuagint because as far as we know, they’re writing in Greek, and it is a Greek version of the Scriptures they and their audience are familiar with. Now some NT authors are adventurous and provide a quotation or allusion that differs from the Septuagint text: even habitual Septuagint users will at times use a different translation of the passage if that proved to be more suitable to their needs.
This is actually a beef I have with people who simply say that “Jesus used the Septuagint.” Well, that’s an oversimplification. We don’t have tapes or videos of Jesus speaking or preaching or books authored by Him personally, so to be honest, we don’t know at all how He cited Scripture or in what language He read them. What we only have is the writings of people who knew Jesus, and people who knew people who knew Jesus, written in a language different from what Jesus would have spoken in everyday life (most likely Aramaic) - so that’s two to three degrees of separation. It’s more accurate to say that “
the NT authors often used the Septuagint when they needed to cite or allude to Scripture.”