Egress window installation
 

In order to enclose our lanai we need to enlarge one of the bedroom windows to egress size. It’s a cinder block wall and we were wondering if anyone else has had this done and, if do, how much it cost as well as any contractor recommendations.

It sounds like you need to widen the window opening, my understanding is that you will need a new lintel beam, as well as cutting the new opening and the window. A neighbor had something similar done (window converted to doorway) as part of a larger project, I believe this change was several thousand dollars several years ago. Original neighbor has passed away, so I cannot verify my memory.
You should be able to contact some of the local construction firm to get a rough idea of the cost. I would also ask the local buidling department if an alteration of the new birdcage (additional door?) would work.