Also subject to the check-rule as usual.
This means that the players need to ensure the safety of both the king and the queen at once.
Otherwise the rules of chess apply, with the following differences:
Castling with the king, as in chess.
Castling with the queen, similarly.
Castling with both.
The queen cannot move into check, so it cannot give check to the opposing queen, because that would put itself into check as well. It can give only distant checks to the king.
The Scholar's mate does not work, because the white queen would move into check (from the black king).
White won, because both the king and the queen are in check (fork), and there is no way to save both.
Black won. The white king is in check, and cannot move away from the back rank, because then the white queen would be exposed to the check.
White should not promote to a queen, because then the black knight could fork the two queens, and then White would lose immediately.