Well yeah, Picard is generally pictured as man well developed in different aspects of "highbrow" classical culture. In Darmok, he can recite ancient Sumerian legends without any preparation. At the end of that episode, he's reading ancient Greek (at least, that's how I interpreted it). He lectures Data about Shakespeare. He's shown to be working on his own proof of Fermat's last theorem in another. He paints, he listens to classical music, has a love for archaeology and history-- to name just a few of his pastimes.
So I don't think it would too out-of-character to have him speak Klingon to a rather high standard, too.