I could be wrong, but since Firebase is a JSON data store and not a server, you probably won't be able to make POST requests via the Ionic Push API. Read the Sending Push doc for reference about how to use the Push API.
That means you might need to build a Node server hosted it on Heroku that's wired up with your Ionic app in order to send POST requests on a user triggered action.
I have been using Zapier.com, which is a service that has recipes for APIs. So if you connect your Firebase account, you might be able to use Parse or Pushwoosh or another push notification service to trigger push notifications. I haven't been able to successfully do this myself, but it might be worth exploration.