Is having a fixed number of edges the only criterion, or do you also need them to be connected graphs?
If you only want the subgraphs with a given number of edges, regardless of whether they are connected, you can use Subsets
SeedRandom[42];
g = RandomGraph[{8, 12}];
edgelists = Subsets[EdgeList[g],{5}];
subgraphs = Subgraph[g, #] & /@ edgelists;
Length@edgelists
(* 792 *)
