This solves your problem?
julia> result = collect(enumerate(eachrow(graph)))
6-element Vector{Tuple{Int64, SubArray{Int64, 1, Matrix{Int64}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}}, true}}}:
(1, [0, 2, 4, 0, 0, 0])
(2, [0, 0, 1, 7, 0, 0])
(3, [0, 0, 0, 0, 3, 0])
(4, [0, 0, 0, 0, 0, 1])
(5, [0, 0, 0, 2, 0, 5])
(6, [0, 0, 0, 0, 0, 0])