You can not drop items from the quest item section of your inventory. There are items involved in quests that don't get put in that section, though (ie the 2 scrolls you find in the fjords for the treasure hunters quest get put in with the books and formula, etc. I never checked if you had to keep these with you, or if finding them was enough to complete the quest).
What items do you need, that you think you may have dropped?
