the true quests from previous version of DF unfortunately have been all removed from DF2014, all you have now is a rumour system and none of the NPC seems to really care about what they're telling you anyways..
To get those rumour things :
- wait until someone (ally or random NPC) tell you about a problem, you reply to him and can ask him where is the problem located
- ask an NPC for problems and he will tell various kind of problems, reply with bringing the subject on a specific one and you'll be able to ask where it is.
- if you're a heartperson you can ask your lord/lady about heartperson duty, sometime instead of telling you there's no problem, they may give you a real quest (good luck for making it works)
But most of those are not quests with an ending in the end, very likely because it's not really completely implemented and sometime even broken as if you get rid of the "problem", whatever it is, those NPC will most likely reply at 99% of chance "it was inevitable" and that will be it (and will probably re-tell you about the same problem later, ignoring that you already told them you got rid of it)
Very rarely an NPC will be actually happy the "problem" is resolved (like in the case of children being freed and brought back to the parent), and sometime they will even accuse you of being a murderer for solving the problem they pointed you to...