Before you can use the PostgreSQL plugin for Qt you will have to build it. Building the plugin has nothing to do with your project or its PRO file and your project does not need to link with PostgreSQL libraries directly. You only need to link to PostgreSQL libraries directly if your code uses the PostgreSQL API directly, i.e. not through the Qt Sql libraries.
Open Qt Assistant and read the page Sql Database Drivers and particularly the bit about building the PostgreSQL driver on Mac, Linux, or Windows. Just substitute the correct PostgreSQL locations for your machine in the build commands. Once the plugin is built and installed QSqlDatabase::drivers() should show QPSQL.
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