Are quest cards considered support cards for the purpose of using cards to destroy them?

No. They are Quest cards, a different type. I can't think of anything off the top of my head that will destroy them.

So far, there is no way of removing Quest cards as Entropy42 correctly states. Who knows in the future? The only ways to stop Questing is to cancel it as it is played, or pick on the unit questing there. Attacking the Quest Zone only works if the defending player chooses to defend with the questing unit, or, you have a card that 'forces' a unit to defend. Other cards can put damage on, or move away a questing unit. However, once in play, a Quest card is always a potential threat depending on its abilities. More Quests appear every new deck, but so far, no Quest removal. Cheers!

At the moment. I'm pretty sure the game is in the right place with quest cards. Most of them aren't powerful enough to warrant destroying the quest card outright as just using one of the many unit manipulation cards instead to knock the unit off the quest. Plus, since quests can't be removed, they are a loyalty symbol that can't be removed from the table once played.....

I agree, quest cards are in no way jeapordizing the balance of the game at this point. If anything, they are in need of a little buffing...