Ok, I've looked around for a solution but nothing works yet.
I'm playing on the 360 and the quest won't start because the guy who is supposed to talk to Elisif to get things started just stands in the corner and never moves. Falk Firebeard has nothing to say on the issue, so I am stuck and can't start the quest. I've heard that one solution was doing the first quest for the Bards College and that might help get things jump started with this quest, but I went through the entire quest with the burning of King Olaf and that did not help.

Suggestions?

Are you doing the civil war questline?

I had this issue where the quest wouldn't start a discovered that the Civil War line can block the start of this quest. Completing the civil War cleared whatever the block was and allowed me to start the quest.

If you haven't started the Civil War, then this is probably not the problem.

Can you talk to the guy from Dragonbridge asking for help?

Exiting and re-entering the Blue Palace does nothing?

How about sleep a night at The Winking Skeever and try the next day?

I have not started the civil war quests at all. I am at the point in the main quest where I am supposed to call a summit between the Stormcloaks and the Empire at High Hrothgar - but I have not spoken to either group yet about attending.

Yes I have tried waiting and coming back into the Blue Palace.

Talking to the guy who is supposed to petition Elisif for help does not do anything. He just tells me to obey the law while in Dragon Bridge.

I'm at a loss...

How long have you waited for him to move? Not the "rest" waiting, but actual waiting where your camera goes into "Circular Pan" mode?