Your Uncle Dodge!
If it is white's move, the queen kills the black rook. If it is black's turn, he kills the white queen and places the white king in check. The game is not over in either case. A king not in immediate jeopardy, as well as being able to move to a safe square if in check, is not a checkmate. You two should practice mating a solo king with just a king and rook (no other pieces on the board). Not as easy as it looks, and a fundamental part of chess.
You need a teacher. You do not know squat about Chess play.
If it's white's turn, and he's going to move the queen, then he should do queen-takes-rook. He can't move the queen in a way that would put his king in check.
The White Queen is blocking the Black Rook from checking the White King. So it cannot move in such a way as to unblock the check on its own King. Imagine that centuries ago, when Chess was first invented, that the game would end with actual capture of the enemy King. But if you attacked the enemy King then the enemy King would always move out of the way if it could. So the only way to win was to attack the enemy King, and at the same time attack every square it could move to, and also be sure that the piece you are attacking with can't be taken and can't be blocked. And then as the game evolved people decided to end the game when the King was under attack and couldn't escape the attack, as the last move in which the King would actually be taken would be a mere formality. So if you think in this way, yes moving the White Queen may attack the Black King, but on the next move Black Rook takes the White King. Then the game is lost, and White Queen can't move any more. In this position the question of who will win depends on whose turn it is. If it's White to play then his best move is Queen takes Black Rook. If it's Black to play then he plays Rook takes White Queen which puts White King in check. Either way it's then a fairly easy win for the side with the Queen or the side with a Rook.
Thomas S: First a terminology correction
First a terminology correction: The piece in the upper-left corner is called the Black King, the next piece over to the right is the Black Rook, the piece below that is the White Queen, and the piece furthest down the board is the White King. In this case the White Queen only has three legal options. The first is to move forward and take the Black Rook. The second is to move forward one space and be adjacent to the Black Rook. The last is to move backwards one space to be adjacent to the White King. The White Queen has *no* other legal moves. Moving the Queen to the left or right at all means the Black Rook would have the White King in Check, and that is not a legal move. By the rules of chess you are never allowed to do something that ends up with your King in Check afterwards. Even if the turn puts your opponent's King in Checkmate, if it causes you to go into Check when you do it, it isn't a legal turn. The White Queen is technically not required to take the Black Rook, but it would be foolish to do otherwise. You also don't have to move the White Queen at all. Technically any movement the White King could make would also be legal. It wouldn't be the best Idea to do so, but it is legal. -- If it is Black's turn, they can technically move either of their pieces in any way that is legal for them. It would be foolish of Black to do anything but take the White Queen though. HOWEVER, if Black were to do this, it would not be an automatic win for Black. To win chess you need to put your opponent into Checkmate. This means you must put the opponent's King into a position where there are no legal moves, while also putting that King in check. Doing this with a single Rook and King is actually very difficult, though not impossible. Honestly, if this would happen to me I would end up calling it a Draw.