How to deal with your opponent's queen if your opponent rejects any trades? If my opponent brings out his/her queen early in the game, I usually attack it with my own queen. The opponent can either accept the trade or reject it. If he/she accepts, I have no questions at all. The game continues, and I am happy.

The question is: what if the opponent refuses to make any trades after I suggest the trade of queens? E. g. I am playing white; my opponent, black. He brings out the queen, I attack it. Then he simply moves the queen to a safe square. Quite 
often it results in me being checkmated in about 10 next moves, or I lose a piece.

So if the opponent constantly refuses to trade queens in the opening, what should I do? Keep attacking the queen (possibly searching for forks/pins/skewers?), or develop my minor pieces, or try to defend my king? Thanks.
There is no need for an early queen trade. Just develop your pieces, if possible by attacking the queen. Here is an example from the Wayward Queen Attack also known as Patzer Opening.

[fen ""]

1. e4 e5 2. Qh5 Nc6 3. Bc4 g6 4. Qf3 Nf6 5. Ne2 Bg7 6. Nbc3 O-O 7. d3 d6



2.Qh5 threatens to take e5, so we defend it by developing a piece: Nc6 (d6 is also possible).  
3.Bc4 threatens to mate on f7, so we defend it by attacking the queen with g6, creating a square for our bishop as a side effect.
4.Qf3 renews the threat, so we develop another piece, Nf6 and block the queen's access to f7.
Then we develop our bishop, castle and get a nice game with a development advantage.


The best punishment for an early queen development is not a queen trade, but using this queen to develop your minor pieces with a tempo: by forcing the queen to move again and again.

Update

In the example you gave in the comment, you play just the same way: Develop your other knight, get castling in, and have a good game. There is a reason why you don't see this line in games by players with a decent rating.

[fen ""]

1. e4 c6 2. Nf3 Qa5 3. Nc3 d5 4. Be2 dxe4 5. Nxe4 Bf5 6. Ng3 Bg6 7. O-O Nd7 8. d4


First let me say that 2.Nf3 is not the best move against the Caro-Kann. d4, Nc3 or even c4 are better and therefore more common. See my Caro-Kann study for more ideas.

After 2…Qa5 you just play Nc3 which develops a piece and unpins the d-pawn. Then you clear the f1 square by developing yet another piece, Be2, castle as soon as possible, and then play d4.
Black is at this point seriously behind in development: the entire kingside is still in the default position while you have already some very annoying moves like Nh4,  c4, or Bf4. Your queen on d1 is keeping everything together. It's not in the way of any other piece and much better than Black's queen – and you didn't even move it!

So, again: stay calm, develop your pieces, castle, and then question your opponent's position with your own minor pieces.