I am trying to do web push notifications in Chrome without using GCM/FCM. Is it possible? I'm not able to find examples on how to use a different push service.
No, it is not possible to use another push service.
In Firefox, you can do it by modifying the dom.push.serverURL
preference, but obviously you'd need privileged access to alter the value of the pref.
There are third-party services that you can use to implement push notifications, but they will use the Web Push API under the hood (so Autopush on Firefox, GCM/FCM on Chrome).