You can if the opposing player's pieces is in the way of one of yours.
Only in certain circumstances such as having your king in check with no squares to move to and the only way to stop the check is to capture the piece delivering the check.
The objective of chess is to checkmate the king, where the king is in check by a piece and it cannot block the check, move to another square, or capture the piece checking the king.
No, you cannot castle out of check in chess.
No, the king cannot kill the queen in a game of chess.
No, you are not allowed to castle out of check in chess.
No, you cannot castle when your king is in check in chess.
No, you cannot castle through a check in chess.
No, you cannot castle through check in chess.
No, it is not possible to castle out of check in a game of chess.
No, a king cannot check another king in chess.
No, you cannot castle in chess if your king is currently in check.
No, you cannot check a king with a king in a game of chess.