Spock speaks many languages. I know this because he is Minsk and the very definition of that means he at least speaks 3 or 4 languages. It is not broadcast in the show though. It's just another one of those things that you have to figure out for yourself or draw your own conclusions on. We know he speaks English. And there is evidence he speaks Klingon too.
And, he speaks Vulcan. So, there are most likely a vast number of languages he also speaks. Like many things in Star Trek, we just have to put our own opinion in the equation and go with that. I would be sure that being such a well-versed one, he might speak up to ten languages. But...who knows for sure? Unless we are told in the show, we can never know for sure.