Thank you for your help so far.
My SQL_BSP.pro now looks like this:
Qt Code: 
  1. QT       += core gui
  2. QT += sql
  3.  
  4. TARGET = SQL_BSP
  5. TEMPLATE = app
  6.  
  7. SOURCES += main.cpp\
  8.         mainwindow.cpp
  9.  
  10. HEADERS  += mainwindow.h
  11.  
  12. FORMS    += mainwindow.ui
  13.  
  14. OTHER_FILES += \
  15.     libpq.lib
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I tried to get access to the sql db like this:
Qt Code: 
  1. #include <QtSql>
  2. #include <QSqlError>
  3. #include <QSqlDriver>
  4. #include <QSqlDriverPlugin>
  5.  
  6.     QSqlDatabase dbAlex = QSqlDatabase::addDatabase("QPSQL");
  7.     dbAlex.setHostName("localhost");
  8.     dbAlex.setPort(5433);
  9.     dbAlex.setUserName("postgres");
  10.     dbAlex.setPassword("***");
  11.     dbAlex.setDatabaseName("QT");
  12.     dbAlex.open("postgres", "***");
  13.  
  14.     if (dbAlex.isOpen() == true)
  15.         m_sConnState = "true";
  16.     else
  17.         m_sConnState = "false";
  18.  
  19.     QSqlError dbError = dbAlex.lastError();
  20.     m_sConnState = dbError.text();
  21.     ui->lineEdit->setText(m_sConnState);
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The message in my lineEdit now is "driver not loaded".

I now, I habe to work with QSqlDriver and QSqlDriverPlugin, but I have no idea how.