Can Munchkin Quest be used for classic Munchkin?
Short answer: No

Longer explanation:

They are totally different games. Munchkin is a card game. Munchkin Quest is a board game. They share:


A theme: Kill the monsters, take their stuff, stab your buddy - parodying power-gaming RPGers of a certain stripe
An artist: John Kovalic 
A goal: Reach level 10 first
A Publisher: Steve Jackson Games
An official website: http://www.worldofmunchkin.com/


But the materials in MQ will not allow you to play Munchkin. And no amount of Munchkin sets will allow you to play MQ.