Do you mean the regular ARR 'quest complete' fanfare, or the track that plays during the cutscene just before (the cheesy elevator muzac)? If it's the later, then that's easy: it's Supply and Demand by Nobuo Uematsu, it's the old FFXIV 1.0 Market Ward track (it's on the Before Meteor OST Bluray album). I'm not sure if it has an Orchestrion roll yet or not, although something keeps telling me it does, I have no proof on that at the moment.

The ARR 'quest complete' track unfortunately does not have an Orchestrion roll, or even a title as far as I'm aware (it's not even on the ARR OST Bluray) I'm sorry to say.