I am visiting Yilan, Taipei in the 1st week of June and wonder if there would be typhoon or landslide risks in this area.

Currently, it is hard to tell the exact weather information, because it is so early.
June is the typhoon season, but it seldom happens, so if you are lucky enough you will not meet one. Besides, if it is not a serious typhoon, it will not affect the traveling. For the landslide, if there is no heavy and long-lasting rain, no landslide will happen. So take it easy.