When the Cloak of Displacement is used, the Quest tokens dont move with the heroes. The Quest tokens drop on the space where the heroes where staying. Both of them.
Can you collect the Quest token with an acitve hero without using a movement point, like you could if it would be an Exploration token?
What is when the space where the hero is staying isnt empty?
I dont have the scenario book online, maybe you need to end your turn on it or the space must be free?
http://arcadiaquest.com/dl/AQ-Rulebook.pdf page 25
Cloak of Displacement
Permanent
Gear
At the start of your turn, you may switch places
between this Hero and an ally (drop all Quest tokens).