There is no functionality for MATLAB's graph class that would compute the bridges / cut edges.
If performance is not a main concern, you could go by definition: For each edge, try removing that edge and check shortestpath to see if there is still a connection between those nodes. If no, that edge is a bridge. You'd need to look at rmedge and shortestpath functions for this. An alternative to calling shortestpath would be to call conncomp and check if the number of connected components has increased - this might be slightly more expensive.
One more thing you could do to limit the amount of edges that are candidates for being a bridge is to call biconncomp. The second output contains the articulation points (or cut vertices), and according to Wikipedia, "The two endpoints of a bridge are articulation vertices unless they have a degree of 1, although it may also be possible for a non-bridge edge to have two articulation vertices as endpoints." So if the two endpoints of an edge aren't articulation points, we know ahead of time that this edge cannot be a bridge.