Quest - Create Quest
Create a variable with the type Quest.
Create the action, "Set Variable" and set the newly created variable to the last created quest.
Quest - Create Quest Requirement (requirement for the quest, set a description)
To update the quest:
Update the quest message: Quest - Display to Player Group - Player 1 (Red) the Quest Update message: Some random quest description here...
(make it for owner of triggering unit instead)
Now all you have to do is create a second trigger for the complete condition. For example create a trigger that when a unit dies and is equal to the unit that you need to kill for the quest, just use this:
Quest - Display to Player Group - Player 1 (Red) the Quest Completed message: |cffffcc00QUEST COM
Quest - Mark Your_quest_variable as Completed
yet again change it to owner of triggering unit...