ОК, я нашел правильный путь:
QEvent *movePress= new QKeyEvent(QEvent::KeyPress, Qt::Key_Left, Qt::NoModifier);
QApplication::sendEvent (focusWidget, movePress);
QEvent *moveRelease= new QKeyEvent(QEvent::KeyRelease, Qt::Key_Left, Qt::NoModifier);
QApplication::sendEvent(focusWidget,moveRelease);