Unobtrusive quest is general quest design direction that makes quests less requiring of interaction and generally more integrated into level design.
Any quest may be made "unobtrusive" or at least more integrated.

Key ideas:
1.Quest doesn't have quest taker NPC or quest reward NPC. NPCs interaction is absent or only provides hints to quests nature(NPCs don't advertises anything about the quest).
2. Quest appears in quest log/queue when player activates it, either by interacting with environment or reaching a specific area.
3.Quest doesn't include specific, concrete tasks. Interaction with environment, puzzles and instanced content, mini-games(integrated into theme of quest) feature prominently.
4.Discovery/Exploration/Unlocking new content is central.
5.Quest doesn't present choices to player or any dialogs.
6.Quest reward is integrated with completion of final part(e.g. access to a treasure room).
7.Quest log adds quest-related progress, but no complete idea of level of progress.