1-based position of an edge in EdgeList (TwoWayRule or UndirectedEdge spec)
EdgeIndex[g, e]
EdgeIndex[{v -> w, …}, …]
g = CompleteGraph[4, DirectedEdges -> True, EdgeStyle -> Arrowheads[Small]]; Table[EdgeList[g][[EdgeIndex[g, e]]] === e, {e, EdgeList[g]}]
→ {True, True, True, True, True, True}