In the game chess can a king move to a square next to another king?

If by moving up to a square, you mean an adjacent square, yes, as long as the square is not threatened by any opposing piece. This is the only way a King can attack another piece.

A King cannot enter any square that is attacked by an opposing piece, i.e. place himself in check. He also cannot castle across an attacked square (king's bishop 1-f1, or queen 1-d1), but it does not matter if the rook or queen's knight squares are attacked, because he does not cross or land on those squares. He cannot castle out of check.