O is basically just a circle, so unlike with C/G, the visual similarity with Q is trivial. You could equally wonder if C being O with a chunk taken out has to do with anything.
Q and O derive from two different Phoenician letters, qop and ayin; back then, they actually did have something in common since both were "throaty" consonants; however, since the shape of the letters is ultimately hieroglyphic, the similarity is either accidental or — if we credit the theory that the hieroglyphic meaning of qop was "eye of a needle" — based on a real-world visual similarity, since ayin/O represented a (literal) eye.