First of all, we can't be certain that Jesus did have a conversation with Pontius Pliate. It is recorded that he did in the Gospel of John, namely in John 18, which contains the famous exchange between the two culminating in Pilate's ambiguous question translated in the Latin Vulgate as "Quid est veritas?" -- "What is truth?"
But aside from the fact that Gospel is a religious document which is generally assumed to be the latest of the canonical gospels (the others do not record such an extensive conversation between the two as John) -- written circa 100 AD (Lincoln, 2005) -- the text presents internal problems as well if we are to take it as any kind of historical document. Namely, it is unclear how the writer of the Gospel would have been privy to any conversation between Jesus and Pilate, given that it is said that it took place in the praetorium -- Pilate's tent, presumably a private place where none of Jesus's disciples would have been present to hear the conversation. So unless Pilate himself related the conversation to the author or one of his sources later on, it is difficult to see how the existence, let alone details, of this conversation could have been accurately relayed in the Gospel.
Having said that, if we assume that Jesus did converse with Pilate -- and all four canonical Gospels do at least have Pilate ask him at trial whether he is the King of the Jews and Jesus reply "You say so," though unlike John, Matthew for example has Jesus remain silent beyond this -- it is not certain that the conversation would have been in Latin. During this time and in this region of the Roman Empire, the lingua franca (language commonly spoken between people who do not share a first language) was Koine Greek -- the same language the New Testament is written in (Ostler, 2005). It is highly likely that Pilate spoke Greek, as not only was this the case, Greek was also the language not only spoken but preferred by many Romans. Indeed it was the second official language of Rome, alongside Latin.
Jesus, though probably a man from a less educated background who predominantly spoke Aramaic rather than Greek (Myers, 1987), would likely have at least known enough Koine Greek to converse briefly with Pilate, since he had ample opportunity to pick up on the language due to its widespread usage throughout Galilee as the lingua franca. Indeed, it is recorded elsewhere in the Gospels (i.e. Mark 7) that Jesus spoke to Greeks, which presumably would have been in the Koine Greek language.
In any case, if Pilate and Jesus did speak Koine Greek enough to converse in its intended form as a lingua franca, there would be no need for Jesus to know any Latin to speak with Pilate, or Pilate to know any Aramaic to speak to Jesus. If Jesus was however only partly fluent in Greek, that could explain why his conversation with Pilate was so sparse in three of the four canonical Gospels, and the details of the more elaborate conversation in Pilate's praetorium written in the later John may have been invented. Alternatively, of course, there is always the possibility that the two spoke through an interpreter of some kind.
So, needless to say, it would not have been necessary that Jesus learned Latin in "Bethlehem schools" -- and it should be noted that, while the two Gospels with a birth narrative (Matthew and Luke) do describe Jesus as being born in Bethlehem, all four describe him as having been raised in Nazareth -- whether or not the language was taught at such schools, and whether or not Jesus attended such schools or even was the son of a carpenter. There are multiple other plausible explanations for how Jesus may have conversed with Pilate, whether or not he actually did and to what extent he actually did, that do not require him to know Latin.
Sources:
The Gospels of Matthew, Mark, Luke and John. (New American Bible Revised Edition, 2011, verses linked in comment).
Lincoln, Andrew (2005). Gospel According to St John: Black's New Testament Commentaries.
Ostler, Nicholas (2005). Empires of the Word.
Myers, Allen C. (1987). The Eerdmans Bible Dictionary.