I see in the FAQ that "if players do not commit any characters to a quest . . . the threat in the staging area still counts against the players." Does this mean that not questing is automatically the same as failing a quest, with threat going up each turn by the amount in the staging area?

It is not "automatically" the same as failing. It is possible to have nothing in the staging area during the quest resolution step, which means there would be a threat of 0. If you did not commit any characters to the quest then you have willpower of 0, that is a tie and that is considered neither a successful quest, nor an unsuccessful quest.

Although it is not an automatic fail, it is probable. Because if the threat in the staging area is greater than your willpower your characters committed to the quest then you have "unsuccessfully quested". Even if the players do no commit any characters to a quest, the threat in the staging area still counts against the players. At that point each player must raise their threat dial by the amount of the threat in the staging area.