I have encountered a trapdoor that I can't open, as well. How does one take pictures with DOSBox? The one I am referring to is in Garenor Hall of Daggerfall.
Garenor Hall is a large, 4 module dungeon. There is a trap door in the southern most module at the bottom of a short shaft. That trapdoor can't be opened (I think). However, it doesn't matter. If there is anything beneath it (which I doubt) your quest object is not there. This can be proved by utilizing the [ or ] key in cheatmode.
There is another trap door in the center module at the bottom of the pit in the throne room. (Your quest object could possibly be here). It is opened by a lever behind a magically locked door on the south side of the pit.
There is another possible location of your quest object in this same throne room. INSIDE one of the statues along the wall (outside the pit, but on the same level). This statue can be shifted (revealing the quest object) by throwing a small lever on the back side of one of the thrones on the upper level.