It’s useful to also show the errors you get.
In that case I think you need to do
@constraint(model, x[Edge(1 => 2), 1] == 1)
As far as I know JuMP cannot understand Graphs.jl so you have to give it the numbers. Use get_prop! from MetaGraphs.jl for that.
Probably a good resource is GitHub - JuliaGraphs/GraphsOptim.jl: A package for graph optimization algorithms that rely on mathematical programming.
Have a look at the source code to see how they define models.