The term was apparently coined by Yu Suzuki, the creator of Shenmue.
Other games, such as Shenmue (Sega, 1999), the game whose director Yu Suzuki coined the terms "Quick-Time Event" and "QTE," are not so forgiving. Failure at a QTE will result in player death and a game over condition. src
Shenmue, which came out in 1999, was not the first game to include QTEs but it was one of the first to include them in the form that we recognize today, of a cutscene with buttons flashing on-screen to be pressed. (Earlier games like Dragon's Lair didn't actually tell you what key to press, you just had to figure it out.)
Originally they were called "quick timer events", meaning the events were driven by a timer, as opposed to the games usual mechanics, and were intended to be "quick". This is the term that appears in Shenmue's manual, for example. For whatever reason, the slightly shorter term is what eventually became standard.