The Qing dynasty era ended when Taiwan was ceded to Japan 日本 with the signing of the Treaty of Shimonoseki at the conclusion of the First Sino-Japanese War in 1895. The Japanese invaded Taiwan that same year, wiping out all resistance and ushering in half a century of colonial rule.
For the purposes of this project I have chosen the neutral phrase Japanese Colonial Era 日治時代 to describe this time period but Chinese nationalists might prefer to describe it as an "occupation". The Japanese heavily developed and modernized Taiwan but not without cost to the local populace, particularly as the war effort ramped up in the 1930s. Following their defeat in World War II the Japanese were expelled from Taiwan and a new set of masters moved in. Thus began the KMT authoritarian era.