Scalable board games, including Five in a Row (or gomoku) and weiqi (or go), are generalized so that they can be played on or by quantum computers. We adopt three principles for the generalization: the first two are to ensure that the games are compatible with quantum computer and the third is to ensure that the standard classical games are the special cases. We demonstrate how to construct basic quantum moves and use them to set up quantum games. There are three different schemes to play the quantized games: one quantum computer with another quantum computer (QwQ), two classical computer playing with each other on one quantum computer (CQC), and one classical computer with another classical computer(CwC). We illustrate these results with the games of Five in a Row and weiqi.