I mean Simon Says as in the game flashes an X on the screen, you press X, and your character doesn't die. Ocarina minigames were more like just plain
Simon.
The FF8 example involved pressing a button for a punch, kick, or headbutt as you (Squall) and an enemy grunt tried to knock each other off a rope dangling from some kind of flying machine. If you didn't win, you'd fall to your death and get sent back to try again. There was a similar event later on involving a dragon that was supposed to be a movie prop but -- oops! -- turned out to be real.
In the "flash a button on the screen that the player has to hit" category, the intro of FF9 involved exactly this in a fake sword fight in a play. Random buttons to press briefly flashed, and you had mere moments to press them. How many you nailed related to how much the crowd liked your performance, which I think had a monetary reward.
This sort of gameplay changeup with rewards goes back at least as far as FF6, where you won different items based on your dinner conversation in one scene, but button presses and timing started in FF7. In particular, I recall a scene in which you had to march in step in a parade and do some tricks with your rifle.
Maybe I'm being too loose with the definition of a QTE, but Square has a history of handling scenes that wouldn't work in the game's combat engine with timing, pattern recognition, or simple button pushing challenges. Combine that with Square's older attempts to merge gameplay with FMVs, and full blown QTEs are an obvious direction for them to go, if they haven't already. I haven't played any FF since FF9.