I don't feel like there are currently enough things to do outside of customs for this to be "fun." It would quickly become a chore that you only do because you want some more ip or key fragments or whatever the prize is. An example of a past quest gone wrong would be the icons given back in ascension, where people would just quit or give up the second the enemy had an ascended because you had to not let them get any ascensions for one of the icons.
Simply making quests be "win X # of games" also puts more unneeded pressure to win a normal game. People will be more predisposed to dodging when you have that weird bot lane duo than they already are relegating far off meta playstyles into customs or friends only games. Also anything that involved ARAM would lead to a lot more people dodging.
The only "good" quests I can think of would be play x number of games with a certain role or x number of games of a certain game mode, regardless of win or loss. This once again, would end up becoming a chore for some people, but at least it would have fewer potential consequences for casual games.