When only queens and kings are left on the board, the game is usually drawn.
Against good defence there's no way to force the win in this type of ending.
This being said, you'll have to be on the lookout for special circumstances in which a win may be achieved..
Whoever moves first here wins.
The x-ray attack 1. Qd4+ (or 1...Qe8+) wins the opposing queen.
Again, whoever moves first wins.
This is another form of the x-ray attack. White wins after 1.Qa5+.
Black to move has to prepare the x-ray attack by playing 1...Qh1+. Now the white king has to leave the first rank and the x-ray is possible. 2.Kd2, Qg2+; winning the queen again.
When the defending king is stuck near a corner, you may find a possibility to mate it.
These kind of positions usually arise after queening a pawn.
The newborn black queen enjoys little freedom and an x-ray attack is in the air.
The black king has to keep the queen safe (usually it's the other way around).
2. Qc4+, Ka3;
The king can't go to b2, because then it will be mated starting with Qb4.
The move to a3 isn't helping either, because of the following sequence.
Again, the king can't leave the queen alone.
If White doesn't find a mating idea, the black king will repeat its moves.
5. Qc5+, Kb2;
And once again the king has to protect the queen (5...Kd1 allows 6.Qg1 mate)
Finally the white queen has acces to b4, starting the inevitable mating sequence.
Black's king has no moves and is stuck in the corner.
The queen has no useful moves either. After every queen move it's mate now (in one, two or three).
If you don't see it immediately, please take some time to check this.
This is another mating pattern, and it's useful to know.
7.... Kb1; 8.Qc2#.
Queen vs queen endings are usually drawn.
There are some exceptions though, like x-ray attacks and mating ideas.
You've seen some examples and I trust you'll use them to your advantage.