The problem is not with Qt. The problem is with your understanding.

The Qt plugin for direct access to PostgreSQL is called PSQL not QSQLITE or QODBC, just as Lykurg suggested. You should prefer the PSQL driver to access PostgreSQL database directly from Qt programs. You can use the ODBC interface to PostgreSQL if there is an ODBC driver for PostgreSQL on your machine and an ODBC data source named dsi64 with the specified credentials.

However, since you have built neither the Qt ODBC plugin nor the Qt PostgreSQL plugin your efforts to connect to a PostreSQL database through either plugin will not get you too far, just "Driver not loaded". Please have read about the supported database types in particular the part about building the plugin for PostgreSQL.