I am on a quest to find a bracelet for the Mage's guild. It's a long quest that has many parts, which is interesting. I've ended up at the Dherkan Plantation which has a room with the four fire brassiers. Clicking three of them is instant death. There is one that has small candles at the base. Clicking that takes me up top on one of the beams. On this beam is a small box that I can't click on.

I am assuming this is for another quest that might be in this dungeon?

I suppose it's possible that might lead to a potential quest location, but with the random and confusing nature of Daggerfall dungeons it's entirely possible that it's something that just goes nowhere, or would lead somewhere if the dungeon module was connected to others in a different way.

(Also, I think you mean "braziers". "Fire brassiere" sounds like an item of clothing I'd think twice before putting on. <_< )

I've clearly never found that little secret. Teleports like that are common enough in dungeons. Usually they just serve as an alternate route somewhere, but sometimes they're the only way to get somewhere. Perhaps the box was meant to be openable, but none of the boxes in dungeons can ever be opened, unlike the ones in houses and shops.

Either way, your quest item will probably be somewhere else.

Quests generally don't work that way in DF. Quest items aren't created in the game world until the quest starts -- which in almost all cases is when you speak to the quest giver asking for work. If you had a few open quests running at the same time, all the quest objects would be clickable.

It's possible that a quest item can be placed next to the chest, however. I do remember cycling through quest positions a few times and coming out on a beam like the one described; don't know if you could teleport to it from nearby lamps, though, because I didn't know to investigate.

Not just "in this dungeon" but rather "in another dungeon using this block". The same sets of rooms and corridors are re-used endlessly, and the teleports are built in to the block, whether they lead to anything useful or not.

I just came across the same block in Medora's Tower on the main quest (and there wasn't anything up there for me either).