I've had a similar issue and from my research and trial and error I can suggest the following solutions using Firebase cloud functions
:
1) You can setup an appengine instance who will read your existing database periodically (or you can mirror your database to firebase) and when the time is right will trigger acloud function as described here
2) You can use a third party service to trigger the cloud function at a specific time. An example for this approach is mentioned here
3) You can also write your own code (maybe javascript/node.js) that can trigger a cloud function based on a certain event (e.g. it can check the changes on your database and then write a firebase database entry to trigger the cloud function)
The solution you will choose depends on many factors but all in all i believe you have a good starting point now.