Could I have the brakes changed? I have a Flying Cat brand electric bike. It is the Style City model.
Currently it has V-Brake SHIMANO AV/AR.
Full details of the bike are here:
http://www.flying-cat.fr/velo-ville-electrique-style-city.php

I am wondering if changing my brakes to progressive brakes (disc brakes) is a possibility. 

This is because while cycling to work someone cut in front of me and I instinctively braked hard. The bike slipped and I fell and have hurt myself. Apparently progressive (disc?) brakes could make my braking safer and more controlled.
I think you have a misconception about disc brakes. You may have heard that (hydraulic especially) disc brakes allow better modulation, which does allow for increased braking control, but if you yank the levers hard, they provide instant, full power braking.

Properly set up V-brakes can provide sufficient modulation to be controllable. Your evidently offer enough power to stop quickly.

Practicing emergency stops on grass is probably a good idea so you have more control of the bike under hard braking.