In my games, I usually try to clear the initial area before I take any quest.

In my last game, during the first turns and before I tried to pick any quest, I saw one quest hut dissapear leaving only the visual effect. It happens in turn 13 which happens at the same time as my Barracks arefinished. But even rushing Barracks and building nothing else, the hut still dissapears by turn 13.

I know that they hut gives "Maggie's Lost Ring" because I checked in one of the replays. You can try here.