Yes, a pawn may capture any piece on the board as long as it is a legal move. One way it could happen is the pawn is blocking lets say a Bishop or a Rook from attacking the opponent's king. The pawn moves to a square where it attacks the Queen. Normally the Queen would either just capture the pawn or move away from it. But if the move of the pawn now places the king in check from that Bishop or Rook (this is called a discovered check) the King must move out of check. If the player now in check has no alternative but to move the King out of check, then once he moves the king, the pawn is free to capture the Queen.
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Yes. It is very rare. It cannot be done alone with a pawn. Other pieces must be "protecting" the pawn. This means there must be, for example, a rook two spaces to the right of it so the pawn can't be taken by the king.
Bottom Line:
The pawn must have protection for this to occur.