Quest only means when you have the quest, then you can obtain the items.
This is excluded for if you are getting the item to start the quest.
That's what 'quest only' means.
Caits might outright say no on this though, and make it possibly that the drop rate increases for the quest only. I doubt it thought. I believe it is available only while the quest is active.

But thanks for the input. I thought it up while we were discussing something else. Just like, well, excluding "normal" quests and some "party" quests, we should make some just pop up like so.