I would suggest you to create also the concept of Node. Meaning besides you're Edge object you would also have the Node object, better modeling always helps you solve your problems faster. Having a HashMap defined with Node has a key Integer will help in this case, because you can store additional info in the Node instance.
But if you want/need to keep your current structure, in order to know how many edges are leading to a given node you'll have to iterate through the various values of the HashMap entrySet (which based on what you said will be lists of edges) and see which edges have the node you're looking for has a destination node, this information would have to be stored in an auxiliary structure though (such has another map)