Today you’re calling something from your application to get data from Firebase. Some API or web method I expect. Similarly, if you want to send data to Firebase, there will be other calls you can make to push it there directly.
In future, you want MQTT to act as an intermediary (I’m not sure why). Staying with the need to send data to Firebase, you will need something that takes the data from MQTT and calls the firebase API on your behalf. Lots of people have already done this, probably because they were doing some kind of IoT project and used Firebase as their datastore.
I suggest that you have a go at sending data in that way. It will illustrate that there is now an extra component needed in the data path between your application and Firebase, not just MQTT.