Generic Graph using Adjacency matrix - Java
You can simplify the mapping you are using to get from vertex to it's associated object. Instead of using a List&lt;V&gt;, which can only guarantee \$\mathcal{O}(n)\$ lookup time, you could just use an array.
Comments restating what's known should be removed, comments that can be made obsolete through refactoring should prompt that refactoring:

public class GraphAM&lt;V&gt; {
    private Map&lt;V, Integer&gt; vertexToIndex;
    private V[] indexToVertex;

    private int[][] adjacencyMatrix;
    private int index;

vertices (aka vertexToIndex can be eagerly initialized.
all private fields (apart from index) should be marked final.
Instead of passing numVertices to the constructor, you should consider passing a Collection&lt;V&gt;, which removes the need to deal with adding vertices in addEdge.
Finally instead of using a boolean[] for visited, a Set&lt;V&gt; is significantly more obvious. It might come with a minor performance penalty though. 
Finally the search functions do not search for anything... they're just performing an exhaustive traversal and are accordingly useless in their current form...