Agreed, a quest unlocking due to background info is fine and can be fun. Said quest auto-completing at the same time means that it is not really a quest at all since there is zero player input or involvement. You may as well just roll some dice every time the player walks in the door and give them a reward if they roll a 7 or something.
That said, hidden unlockable quests and even hidden rewards can be still be a usable mechanic. In Fallout: New Vegas I unlocked Perks that I didn't even know existed just through regular gameplay - killing tons of mutated ants eventually unlocked a Perk that made me deal more damage to bugs. I didn't set out to get it because I never saw it in the Perk selection at Level Up, I was just killing so many ants because I went to places that had lots of ants. Eventually I killed enough ants to unlocked a second tier of said Perk to do even more bonus damage. Some of these gameplay Perks gave me period completion bars so I knew that I could do things to unlock something, like I think there was a Perk for getting enough headshots with a Pistol but didn't get the notification for the "quest" until I got something like 25/100. If a player went the whole game without using Pistols, or never went for headshots, they would never know the Perk existed. But if that's your play-style then the game will give that notice so the player knows there is some kind of reward if they do that action a bunch more times.
I think the key there was letting the player know what needs to be done to finish the "quest" so they actually have something to do. The ant quest was neat since it was unobtrusive and was just letting me know I got a simple bonus to make doing that repetitive action easier. The Pistol quest was a lot more fun since I suddenly learned that I could get a cool reward and could choose to build up my count toward that reward at any time - since I was primarily using my Pistol on that run I went out of my way to try and get it fast but if I preferred using Rifles instead I could have just ignored it without feeling guilty about missing a serious quest like the ones in town.