There's not much evidence that Jesus was able to read and write, or that he could read and write Greek. Certainly, if that was the case, he managed to fail to record his own message. So I have no idea how you would discern that Jesus used the Septuagint, apart from the much later narratives of the gospels relying on it (in an often humorous way that amuses Jews to no end).
Jesus was entirely literate, which is evident in scripture, and he certainly spoke Greek. Everyone living in the region spoke Greek at that time along with whatever their first language might have been. Greek had become the dominant language following the conquests of Alexander the Great and the spread of Greek culture into Asia Minor, the Near East, and North Africa. As Hellenistic culture became more pervasive, more and more Jews adopted Greek as the international language and Jews, especially those living outside the Holy Land, became less fluent in Hebrew.
By the year 250 B.C., the Hebrew texts had been translated into Greek for this reason, with some newer books added to make up the Septuagint. By the time Jesus was born, Greek was the dominant language even in the Holy Land, and the Septuagint was the bible in use by Jews there as everywhere else. Hebrew was only being used for liturgical purposes. This is also why the New Testament books were all written in Greek.
In addition to Aramaic, Hebrew, and Greek, Jesus would surely have spoken Latin because he engaged with Romans. Actually, we can be certain that he spoke any language he needed because there is no human language unknown to God.
Jesus changed the world forever in a three-year ministry without writing a book. And since most of the common people were illiterate at that time, he reached many more through interacting, preaching, and oral teaching than he would have by writing, so it would not have been as effective. He didn't need to write books anyway. He gave his authority to the apostles and directed them to "teach all nations" what he had taught them. He gave the continued teaching and writing responsibilities to members of his Church.