PHB p 258: Sometimes a quest is spelled out for you at the start of an adventure. . . . You can also, with your DM’s approval, create a quest for your character. Such a quest can tie into your character’s background. For instance, perhaps your mother is the person whose remains lie in the Fortress of the Iron Ring. Quests can also relate to individual goals, such as a ranger searching for a magic bow to wield. Individual quests give you a stake in a campaign’s unfolding story and give your DM ingredients to help develop that story.
DMG pp 103, 122, 125: Design quests so that they have a clear start, a clear goal, and clear consequences. Any quest should provide a ready answer for when the players ask, “What should we do now?” . . . Quests should focus on the story reasons for adventuring, not on the underlying basic actions of the game - killing monsters and acquiring treasure. “Defeat ten encounters of your level” isn’t a quest. It’s a recipe for advancing a level. Completing it is its own reward. “Make Harrows Pass safe for travelers” is a quest, even if the easiest way to accomplish it happens to be defeating ten encounters of the characters’ level. This quest is a story-based goal, and one that has at least the possibility of solution by other means. . . . 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!