Maybe a little late but you can get those shelf brackets (L-shape). Attach to the fence and the bottom of bird house. You can also hang it off a hanging planter bracket. It will move a bit in the wind but otherwise, the birds should be able to get to it just fine and live in it. Just need to be careful of how other wildlife might access the house also, such as cats or squirrels.

I probably wouldn't hang it off the fence, you have no way to protect the inhabitants from predators. The best way is on a pole, with something like a metal baffle around it, far from where anything can jump up to gain access. Like cats or coons.