Normally to complete “Let the Fires Come!” or Stop the Fires! you and some other people would just pick up the quest, put out the fire, and turn it in. If you have alts (especially on the same account) you would usually have to wait a few minutes until you can pick up the quest and do it again.
I have found out that if you have someone complete the quest and then you share the quest to a person who is in the town, they will have their quest completed right away (as long as the quest is still on “cooldown”)
A,B,C,D=My alts (on one wow account)
A, B, C, and D are all in Brill’s Inn before any of this.
-I log on A and do the fire quest with X in Brill.
-X DOES NOT TURN IN THE QUEST
-I log off A and log on B
-X invites B to party and shares the quest
-B’s quest is completed
-B turns quest in
-rinse and repeat for C and D
NOTE: There is also a quest you can pick up after killing the horseman that shows up after you put out the fire. I am not sure what it’s called but pick it up as you log in and out of your alts.
SIDE: I do this so my alts can get free XP and any 70s would get free gold. Not bad for a daily.