Even though the birth of Jesus took place in Bethlehem (which was not the capital of Palestine but the city where King David was born), Jesus was acknowledged by his contemporaries as being from Nazareth, an obscure village in Galilee, the northern region of the country.
This region, like the rest of Palestine, was under Roman rule. There is strong evidence that it had a mixed, somewhat multilingual society, with the Aramaic-speaking Hebrew population being largely predominant. We can see this in John's Gospel (19:20), when it describes the condemnation sign that Pontius Pilate ordered to be placed on the Jesus’ cross:
"This inscription was read by many Jews, for the place where Jesus was crucified was near the city, and it was written in Hebrew, Latin and Greek."
We know through The Jewish War, originally composed in Aramaic for the Hebrews of Babylon, then translated by a Greek translator in the form that reached us, that "the Hebrews" compatriots of Josephus were mostly monolingual: at one point an interpreter is needed to read a correct Greek text.
Hebrew and Aramaic
Roman colonization had increased the mix of populations and languages in the region, and it is almost certain that all inhabitants, to some extent, roughly spoke or understood several languages (in a quite limited way, according to Josephus).
Jesus, like all the children of his time, spoke Aramaic, the Hebrew dialect which was his mother tongue. He also knew Hebrew, the language in which the various sacred books of Judaism had been written, and the liturgical language in the Temple - but not of the country’s synagogues.
Greek and Latin
Jesus had contacts with people speaking Greek, even Latin, the two languages of culture and trade the eastern Mediterranean since the Greek and Roman conquests.
For example in Mark 7:24-30, Jesus goes to the region of Tyre and speaks with a Syrophenician woman. Mark observes that this woman spoke Greek, therefore the conversation she had with Jesus was conducted in Greek.
Another example is found in Mark 12:13-17, where Jesus is seen answering some Pharisees about the duty to pay the tax to Caesar. Palestine used coins bearing the Latin inscription “Divus Augustus”, in the effigy of “the divine Augustus.” Jesus did not ask what the inscription meant, but what name was inscribed on it, a sign that he understood its meaning.
We should not omit to evoke Jesus’ interrogation by Pilate, which could only have taken place in Greek or Latin.
But even though Jesus seems to have spoken and understood several languages (and why wouldn’t he have, as the Son of God?), he must necessarily have adapted his dialect, even his accent, to his listeners. In Galilee he obviously used the dialect and unrefined accent of his countrymen, the same one that betrayed the Apostle Peter hiding in the court of the high priest at the time of Jesus’ trial.