Some quest ideas:
Procuring rare materials (rare gems that could be used to satisfy nobles or for other reasons)
Find powerful items and retrieve them. Items you created in other fortresses that are of sufficient quality (masterwork or artifact) could be learned about somehow, and you could send adventurers after them. If, say, there is an adamantium short sword somewhere in the world that you made, you could maybe retrieve it. Or if you want that artifact table from the last fort for your dining room. Additionally, the game could pre-generate a number of legendary items to use for this.
Find and recruit specific types of dwarves. You could send someone to search for a highly trained weaponsmith, for example, and they might return with a Grand Master or Legendary one.
Return with live captured animals for your zoo. You could send them after polar bears or ogres or harpies or whatever.
What I'm leaning towards here is an adventurer job and either a quests screen that appears when you get the mayor, or a building like a tavern or something that allows you to queue quests as tasks, like other buildings allow you to do.
Then your adventurer enabled dwarves will take quests from the queue and leave the fortress. You would probably need some mechanism to allow them arms and armor or they will likely have little chance of success, so hmm... I actually favor making this a job for the military. In the squads menu you can set squads to "Accept Quests", turned off by defualt. Then the squad will take quests from the queue as mentioned before.
The squad would leave with its equipment and the rng would do some automagic mumbo-jumbo to see if they succeed and if they come back with injuries or casualties. Your chances of success could be affected by some factors: distance to travel, relative danger of your surroundings, relations with surrounding civilizations, and of course skill levels of the dwarves and their equipment.
Returning dwarves would gain experience but have a good chance of being injured or killed.
An added bit of flavor is that caravans could give you bits of news about your heroes: "The humans mention that they have heard that Rak Rakurst has died, and that Melbil Dwarfguy now commands the Anvils of Burnings.