quests and other incentives to playing support are a terrible idea anyways
you end up with a bunch of people picking support because there's a reward or because it's a faster queue time.
but they don't know how to play nor do they care, they just want the reward. so you end up with terrible support players who have no idea what they are doing and they do badly. and that of course cripples your team especially if the enemy support knows and wants to play support. this is functionally the same as having people autofilled into support role since they're basically being forced to play for a reward instead of actually wanting to do supportive things.
the solution to support being unappealing is to actually make it fun and rewarding and satisfying.