Swordmaster of Hoeth states "Battlefield: Cancel all combat damage this turn". Let's say the Swordmasters decide to attack an opponent's quest zone during their turn. Do they lose this ability because they are attacking an opponent's zone other than his battlefield (In other words, is he leaving the battlefield to go to the quest zone?)? Is the same true for the Troll Slayers for the Dwarves?

That effect is just based on the zone the unit is in. He doesn't leave the BZ to go attack a QZ. If you play him (or he gets moved) to your KZ or QZ, he will not have the ability while in that zone.

I Concur. It's the same reason that if the swordmasters were using "protect the Empire" or "Defend Tor Aendris" They would not get their battlefield effect even if they were defending the battlefield with the quests effect.