When speaking about the ordinary language of Jewish people in first century Palestine, the language spoken was Aramaic. Specifically a form of Aramaic known as Jewish Palestinian Aramaic, and we also know from both the New Testament and other historical sources that in Galilee there was a peculiar dialect which those in Judea tended to deride. The Galilean dialect appears to have a major negative association, as though they were backwater and uneducated–also Galilee historically had a large Gentile presence which appears to have negatively affected the Galilean reputation among Judeans.
We might assume that when, for example, the Lord read from the Scroll of Isaiah that He was reading from the Hebrew text; but it’s very possible (even likely?) that it was in Aramaic. Aramaic translations of the Scriptures were common during the 2nd Temple Period of Judaism, and were probably the way most Palestinian Jews (Judean and Galilean) heard the Scriptures, in their own language rather than in Hebrew.
The question of whether the Lord knew Hebrew is complicated; because there are actually several questions we need to ask. The more mundane question would be whether the Lord’s education included learning Hebrew, the Gospel of Luke reminds us that our Lord “grew in wisdom before God and men” which indicates a growth in His understanding and wisdom as a man; at the age of twelve He could hold complex conversations with the learned scholars at the Temple on at least equal footing. So did the Lord, in an academic or semi-academic context learn Hebrew in a similar way that, perhaps, the learned and educated elite may have learned some Hebrew? I think there’s no definitive way to answer that either negatively nor positively–we simply don’t have sufficient evidence to answer that sort of question. But there is also another question: The Lord Jesus was, and is, truly and fully God. As God He knows all things (though, and here is part of the paradox of the Incarnation, there are times when He states He doesn’t know something); such as that He knew the hearts of others–He knew their secret thoughts and intentions. So we would, quite reasonably say, that since He’s God of course He knew Hebrew, He knows every language, He knows what people think before they think it. And yet, we have those moments where we read in the Gospels where the Lord doesn’t know something–He explicitly says only the Father knows the day and hour of the Lord’s return. How can God, who knows everything not know something? Such is the paradox of the Incarnation. So–does this extend to something like knowing Hebrew? Again, we might be asking a question to which we simply do not have an answer.
So the question “Did Jesus know Hebrew?” actually is a very difficult, perhaps even impossible to answer, question.
I’m not, however, sure that the answer to this question, whatever it is, actually affects how we interpret Scripture. What is a central aspect of good biblical exegesis is understanding Jewish thought in the 2nd Temple Period, because it’s impossible to read the Gospels apart from the context of the history of the Gospels. The Gospel Story, the Incarnation, takes place in real history, in a real moment of history, in a specific place and time. So understanding, or trying to understand, the complex contextual factors is tremendously important.
Whether Jesus knew Hebrew is, arguably not very important–but Jesus’ Jewishness, and what the Jewish world of the first century looked like, and the Jewish world within the larger historical context of the Greco-Roman world all matters a great deal. But that matters regardless of whether the Lord ever used the Hebrew language in any meaningful way.