I think you are a little bit confused about database applications. The "server side" is the database management system. The "client side" is the application that connects to the DBMS, stores data in it, performs searches, and displays the results. You do not need to write any code to implement the server, only the client application that talks to it.

You can use almost any SQL-based DBMS with Qt; the Qt installation comes with drivers for most SQL DBMS software. Since you are just learning, I suggest you use Qt's implementation of SQLite. It is built in to Qt and does not require you to install separate DBMS software. You should start by reading the Qt SQL documentation and then studying the Qt SQL examples. You can also find a lot of tutorials and videos online about using Qt SQL.