To be clear, they did try to do this! Hideyoshi and Tokugawa were both interested in establishing Japanese presence on Taiwan, mainly to use it as a trade center where Japanese merchants could more directly acquire Chinese goods. This was interesting historically because it was the Christian daimyo who attempted it;
Arima Harunobu, one of the really interesting figures in Japanese Christian history, tried and failed, setting off a complicated chain of events that eventually led to the persecution of Christians in Japan, and another Christian,
Murayama Toan, was more directly ordered to do so by Tokugawa. So, depending on OP's POD, there are two very different potential effects: if the Japanese successfully colonize Taiwan in the late 1500s, this could allow them some trade with China without European middlemen, greatly reducing European influence and Christianity going forward. If the POD is that Arima or Murayama succeed, though, you could have even more powerful Western daimyo, or a full Christianized Japanese culture developing on Taiwan!