Quick strike action card (event Action card) states you can exhaust a character you control to immediately declare it as an attacker (and resolve its attack) against any ELIGIBLE enemy target.

What determines what eligible enemy target - Do enemies in the staging area count or just those already engaged with the party?

Thanks in advance,

Mike in Pittsburgh, Pa

It means you can attack an enemy that you would normally be able to attack during the combat phase. So you can't attack the staging area (unless you have Dunhere). Ranged characters can attack enemies engaged with other players.