Yes there was a cheese contest! It is written about in chapter 10 of "Anecdotal Recollections of the Congress of Vienna" by Auguste-Louis-Charles de La Garde, an ambassador. Unfortunately, he was not present at the event, and heard about it second hand from another person present at the conference who presumably witnessed it first hand.

The cheese contest occured during the dessert section of a dinner. At first the cheeses on offer during dessert were Stilton, Strachino, Gruyère, Limburg; all of which were invariably supported as the best cheese by men from their country of origin (an English diplomat, Lord Castlereagh, supported Stilton etc.). However, no good contest is complete without a surprise entrant, as a courier brings in missives from France and Brie cheese. Charles Maurice de Talleyrand (probably the most famous French Diplomat at the Congress) throws his hat into the cheese ring stating “Send the despatches to the chancellerie, and bring in the cheeses at once." Apparently, the brie was quite favored as the matter was quickly put to a vote and brie was proclaimed as the "King of Cheeses."

I'm unaware of any other sources of the Cheese contest including any actual first hand accounts of the contest. Unless someone with more knowledge and better sources can jump in, I don't think we can say if brie's victory was unanimous, or if it was a close run thing. Additionally, it is probably not a bad idea to take the story with a grain of salt, as, after all, a Frenchman is telling the tale of a French cheese winning.