Firebase with WordPress instead of SQL?

No you can’t use Firebase as your WordPress database out of the box, and the chances there is a plugin to implement this are astronomically low.

Your best hope is that you can find a plugin that lets you embed Firebase data or interact with Firebase, but that would still need to use MySQL/MariaDB for users/posts/etc, you can’t replace the DB connection with a Firebase connection ( and why would you want to it would be awful for performance ).

Officially, only MySQL/MariaDB is supported, with experimental SQLite database support being worked on.

Sometimes 3rd party plugins exist for other database providers ( notably PostgreSQL ), extremely rare to see, and have compatibility problems with plugins, assuming they work with the latest WordPress. Notably they’re all for SQL type databases and work by trying to adjust the SQL syntax to be compatible with another database.