I am also not entirely sure how much the game has via quests, other things to do and so on. I do gather the game has a lot of things to explore in the world and many items to look at, which is awesome as far as it goes. The players guide does mention something about "puzzles and challenges", but other than the obvious ones directly related to hacknslash, or simply getting experience for the sake of exploration, I'm uncertain what is exactly around.
This is particularly of interest to me since I'm a bit of a questing fan, and while I am always happy to just explore and poke things, and I'm more than happy to treat combat as an optional extra, , I do like tasks to do and to feel that I'm doing something useful.
I think the concept to foreground is "structure". Quest systems in many games exist to provide structures for players to work within, including specific goals to achieve. Whether that's attractive depends on one's personality, I think. Many people thrive in a more structured environment, others chaff against its restrictions. TriadCity is extremely "open" in the sense that you're not forced into specific paths, but instead are left to invent or choose or discover them for yourself. Some people really love this lack of constraint, others find it frustrating because they're not sure how to proceed.
So the narrow answer is "no", there isn't a formal quest mechanism (yet - see below). Yet there are many "puzzles and challenges" which act like quests. Herbalists for example must know where to find the herbs they use, some of which are quite esoteric and difficult to come by. They may be not only difficult to locate, but also dangerous to acquire. They could be guarded by NPCs, or there could be Indiana Jones style puzzles to solve to reach them at all. We often structure these challenges to require groups of players with complementary skills. You may need a Thief to pick the lock to get you into the garden and you may require brawny bodyguards to fight off the guardians. Or you may just pay people to get it for you. We have one veteran Thief character who's highly adept at providing hard-to-get herbs for Herbalists.
We're currently working to provide more of these challenges which can be solved by individuals without groups. And we're discussing with veteran players ways to enhance our existing, quite primitive "treasure hunt" mechanism where NPCs give you mini-quests to find things for them. I don't know where this will go over time. Since you have a strong interest in questing, please jump in! The discussions are in the forums.