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So I don't know the mechanics of these in 4e, but it seems doable.
The only mechanical element to a quest in 4e D&D is that completing a quest earns XP. In practice, that means that the game progresses more quickly when the PCs are engaged on quests.
4e uses "quests" as an organisational concept for play. From the PHB (p 258) and DMG (pp 102, 122):
Most adventures have a goal, something you have to do to complete the adventure successfully. The goal might be a personal one, a cause shared by you and your allies, or a task you have been hired to perform. A goal in an adventure is called a quest.
Quests connect a series of encounters into a meaningful story. The simplest adventures revolve around a single quest. . . . Most adventures are more complex, involving multiple quests.
Quests are the fundamental story framework of an adventure - the reason the characters want to participate in it. They’re the reason an adventure exists, and they indicate what the characters need to do to solve the situation the adventure presents. . . . More complex adventures involve multiple quests, including quests related to individual characters’ goals or quests that conflict with each other, presenting characters with interesting choices about which goals to pursue. . . .
Don’t be shy about letting the players know what their quests are. Give the players an obvious goal, possibly a known villain to go after, and a clear course to get to their destination. That avoids searching for the fun - aimless wandering, arguing about trivial choices, and staring across the table because the players don’t know what to do next. . . . Thinking in terms of quests helps focus the adventure solidly where it belongs: on the player characters. An adventure isn’t something that can unfold without their involvement. A plot or an event can unfold without the characters’ involvement, but not an adventure. . . .
Completing quests earns rewards for the PCs. These rewards primarily take the form of treasure (both money and items) and experience points, but quests can also have less concrete rewards. Perhaps someone owes them a favor, they’ve earned the respect of an organization that might give them future quests, or they’ve established a contact who can provide them with important information or access.
Both books also speak about player-authored quests. From the PHB (p 258) and DMG (p 103):
Sometimes a quest is spelled out for you at the start of an adventure. . . . Other times, you figure out your quests while adventuring. . . . You can also, with your DM’s approval, create a quest for your character. Such a quest can tie into your character’s background. . . . Quests can also relate to individual goals . . . . Individual quests give you a stake in a campaign’s unfolding story and give your DM ingredients to help develop that story.
You should allow and even encourage players to come up with their own quests that are tied to their individual goals or specific circumstances in the adventure. Evaluate the proposed quest and assign it a level. Remember to say yes as often as possible!
The first paragraph in the second block of quotes (which is from the PHB) illustrates three approaches to establishing quests: GM-dictated ("spelled out . . . at the start of an adventure"), player-established based on the prior events and revealed backstory (an example given on the DMG p 103 is the players deciding what it is that their PCs want to do with the escaped prisoner they have come across), and player-established based on their own PC goals and backstories.
If the GM approves/allows the lattermost sort of quest (which as you can see they are encouraged to do) then they are committed to building adventure elements - setting, encounters, treasure, etc - that will allow that quest to occur. It's basically a formalised system for channelling suggestions from players to the GM who has the actual authority over those bits of the fiction.