Can King take piece end up threatened? In this position black queen defended by pawn checked white king. Is it possible if the white king take black queen and end up threatened by black pawn?
Assume the image was taken from Black's side (otherwise the queen is not protected).

No. The Black queen is protected by the black pawn, and thus can't be taken by the White king. The only possible move is to move the king to the other direction one square.