Not directly. SQL Express (SQLX) is a different beast built on the same engine. You cannot just "upgrade" it to a different edition. You can however have SQLX and SQL Server installed at the same time and many people do as a lot of apps historically used SQL Express (or LocalDB).
You can create databases and develop code using SQLX. Once you are ready to "upgrade" then install the version of SQL Server for the license you have. You can backup and restore the database from SQLX to the new server and change your connection string in your code and you should be good to go.
Note that SQLX doesn't support all the same features as SQL so you'll want to ensure you retest your code. Also note that SQL Developer is free for development use so if all you're waiting on is a license then forget SQLX and just use SQL Developer. This is the closest you can get to the real thing as it is the same product, just with some features disabled. This is for local development and the recommended approach. If you have a separate DB server then install your SQL Server + license on that server and continue to use SQL Dev for local development. The databases can be backup/restored across servers and your connection string just needs to be changed to point to the other instance.