Most aggro decks can beat quest pally. Murloc shaman, tempo rogue, hunter, zoolock, and token druid are all favorable matchups. Control decks suffer in the late game because of the value pally gets from its hero power. If you play a control deck, the only hope is to tech in silences, but it's still not a favorable matchup.
But sometimes even Warrior can beat Quest Paladin.
If the Paladin draws poorly or deathrattle minions are in the bottom the Warrior just clears every deathrattle minion ( like the 0/5 and the 2/2 egg for example). But yeah, generally speaking the easiest matchup should be Warrior.
Also Firebat made yesterday a anti aggro Questpally wich had too little lategame vs Warrior (No Undertaker no Whelps....).
It's a very strange query since Quest Paladin is a tier 2 deck. I've been playing quite a lot on ladder over the last couple of weeks and I've faced only a handful of Quest Paladin decks, I think I beat all of them.
Priest, mage, shaman, aggro warrior, token druid, temp rogue, zoo warlock, basically nearly all meta decks beat Quest Paladin.
I assume the OP plays warrior or hunter, as these are the only two classes to lose against Quest Paladin.