I understand that he would have lived in a Roman controlled area, and I believe that the predominantly spoken languages in these territories did not generally change, but I find it hard to believe that the general people of a long ago conquered land had not picked up the language of their conquerors.
At no point in the gospels does Jesus appear to speak Latin, but also it's not clear whether he is conversing in Aramaic, Greek or Latin but given his background it's unlikely he was educated in Greek or Latin. He would have probably spoken Aramaic and he may have been fully literate in Hebrew given the depth of his debates with the Pharisees and his knowledge of Jewish law and prophecy.
The Romans generally conducted government business in Latin but they did allow people to keep their own language. And in Judea at the time that would have meant Hebrew, Greek, Aramaic and Latin where all used. Greek was actually the most common language among the upper class/intelligentsia and the oldest copies of the gospels we have are written in Greek not Latin. This is a due to alexanders conquest of the region and his successors dynasty's influence on the cultures there.
As far as I am aware there is no evidence that the historical Jesus spoke any language other than Aramaic. Allen Meyers wrote in 1987 in The Eerdmans Bible Dictionary:
It is generally agreed that Aramaic was the common language of Palestine in the first century AD. Jesus and his disciples spoke the Galilean dialect, which was distinguished from that of Jerusalem (Matt. 26:73)."
Hebrew, by contrast, was a language that was mostly exclusive to clerics and religious scholars and as a written language for holy scriptures.
Latin would not have been widely spoken by the Jewish population of Galilee or Jerusalem. An ordinary Jew would not have had any contact with Roman officials who spoke Latin. Any information disseminated to the population would have been translated beforehand. It is very unlikely that Jesus would have had much exposure to Latin, let alone the ability to speak, read or write it.
Tangential question so as not to have a second thread; what were the real names of the apostles? I am fairly certain no one at that time, in that part of the world, was named "Matthew," for example.
Also, quite when did their names get Anglicized like this?
Jesus also frequently visited Phoenicia. In fact, according to the Bible, he ventured to Tyre and Sidon and did miracles there. We do not know if he spoke in Aramaic with the local Phoenicians, or spoke Phoenician.
Aramaic and Phoenician were very similar, however.