Is not possible from random or discover effects directly, but there are some other ways to obtain the quest and then complete it to obtain the reward or obtain the reward directly.
The Priest class has several cards to get the quest from opponent's class.
Since Priest has those cards it means Warlock and Mage with kabal Courier might have access to those cards too.
Following the chain if rogue plays against priest, warlock or mage he can obtain a random card from opponent's class that allows him to steal the quest/reward from opponent's class.
Shaman and Druid can obtain rogue cards from Lotus Agents discover.
Warlock with renounce the darkness can also become one of those classes.
Shaman can obtain a card from opponent's class thanks to Crystalline Oracle coming from Malestrom portal.
And the neutral card Nexus-Champion Saarad can obtain any collectible spell in the game from any class.
And in the end thanks to Elise the Trailblazer and her Un'goro pack every class can obtain other classes cards from Un'goro set except the quest.
The actual cards that can obtain the quest/reward are:
- Mind Vision: 1 mana spell obtain a random card from opponent's hand.
So if the opponent still has the quest in hand it can be obtained with Mind Vision.
- Curious Glimeroot: 3 mana 3/3 minion with battlecry guess and steal a card from your opponent's deck. If your opponent plays a deck with the quest in theory you have a 1/30 chance to steal it with this card, no matter if it's in the deck, the hand or already in play.
Then in the case the opponent mulliganed his quest into his deck there are several cards that can obtain it from opponent's deck.
Crystalline Oracle, Thoughtsteal, Drakonid Operative, Shifting Shade.
Also keep in mind the quest reward that are minions can be placed back in the deck with cards like Madame Goya or Manic Soulcaster, and then be obtained from there.
TL/DR. Every class has ways to get the quest/reward from opponent's class with a more or less complex chain of cards of getting random cards from other classes. Most ways revolve around priest cards like Mind Vision, Curious Glimeroot and steal cards from opponent's deck.