[mtve]

Sorry in advance for not answering right to your question, but you may check sources of ManGOS/TrinityCore/family WoW servers for that.

In short, from what I know, yes, quests are stored in "quest log" fields of character data in server DB, and they are tracked by the clients and checked by the server. Some simple auto quests like "find this item" are not even tracked by server and only stored on completion. Since both client and server have all game data, the client knows about all possible quests and only shows to the player what is appropriate at a current state.