The Dungeon Finder reward Satchel of Exotic Mysteries that usually contains some gold(if not always cant remember) and can be send between factions. I used this to get a horde character started on a server I played alliance on. It is not much gold but it should be enough to get you started if you can run Dungeon Finder a couple of times a week as one of the needed classes.
Also you might get lucky and get a rare item that can be sold or a pet that is only sold on the opposite fraction than you need the gold. That should give you a couple of hundred gold to get started depending on rarity and number of cross fraction traders on server. Just remember to turn autoloot off before opening the satchel to look what was in it.
Other than that the only way as LessPop_MoreFizz said is Paid Fraction Transfer or finding someone on either side to help you with the transfer.