Can you eliminate other persons quest card ?

Generally, no. Except there is one Orc unit - some kind of a Boar dude, can't remember exactly - that if you have him in your battlefield, lets you treat all quest like support cards I think. Then you could Burn it Down or Pillage or something like that. I tried it in a deck a few times but it never seemed to work out. The unit costs 3 and all it's gunna do is sit there for that effect. Just not worth it.

If you're really having difficulty with someone's quests, bring some stuff to directly kill off or otherwise move the units that are on that quest. And just to be a jerk (plus it's more effective and disruptive), if the quest in question needs 3 tokens to fire off for example, let your opponent get to 2 tokens before you smash his/her dreams. Priceless.