Only queenside castling would be feasible: castling kingside would involve moving the king out of the way - too much hassle. You're also relocating the Queen to b1/b8 (QN1), and the rook to c1/c8 (QB1), and any e.g. bishop attacking b1 or c1 would make castling with the Queen illegal.
So not only would castling be very troublesome to carry out, few people would find it useful: give one game of yours where queen-castling would have helped you significantly.
In all likelihood, not much would change.
What would be a theoretical consequence in a chess variant where the Queen was premitted to castle similar to how the King can castle!?