There are several cards that can really disrupt the quest mage.
loatheb is very strong but you have to time it right. If you have a decent board you can drop him and maybe push lethal. Or if you know he’s going to play warp next turn, it’ can buy time. Nerubian Unraveler really fucks them up though. Increased spell cost makes it significantly harder to play a bunch of cheap sells. It also stops a lot of large board clears.
You can also play an anti-secret tech card likee eater of secrets or kezan mystic. Removing or stealing ice block is huge
If you want to go nuts, you could run Azalina soul thief. Once he gets time warp in his hand, you can steal it. You would need some kind of cost reducer though. But you could play his own combo against him if you steal giants and/or vargoth
if at all possible, mill the quest mage. Honestly though it’s an uphill battle to beat them. They have access to so many spell generator cards like ray of frost, and draw like book of spectres.
I think that people is confussing between secret mage and quest mage. I understand that you are referring to quest mage (with giants, flamewaker and so on). Even Shaman, Aggro mech paladin or Cubelock if you can set a wall of taunts.