I remember an episode of The Next Generation where at the end Q gave Data a gift, which was experiencing laughter. But before Q gave the laughter to him, Data thought Q wanted to make him human. Is it possible Q really could have made Data human?
In Hide & Q there is a moment on the bridge when Riker is about to make Data human. Although Data declines the offer, Riker appears to have no doubts that he is capable of making the change.
DATA: No. No, sir.
RIKER: But it's what you've always wanted, Data, to become human.
DATA: Yes, sir, that is true. But I never wanted to compound one illusion with another. It might be real to Q, perhaps even you, sir. But it would not be so to me. Was it not one of the Captain's favourite authors who wrote, This above all, to thine own self be true? Sorry, Commander, I must decline.
Later, in Déjà Q Data warns off Q from making him human, to which Q responds that he'd never curse Data in that way. While Q would be unlikely to admit to not being able to change Data, we can probably assume that it would've been within his power.
Q: Yes, once again I've overstayed my welcome. As a human, I was ill-equipped to thank you, but as myself you have my everlasting gratitude. Until next time. Ah, but before I go, there's a debt I wish to repay to my professor of the humanities. Data, I've decided to give you something very, very special.
DATA: If your intention is to make me human, Q.
Q: No, no, no, no, no, no. I would never curse you by making you human. Think of it as a going away present.