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btns1[i][j]->installEventFilter(this);
...

...
bool MainWindow::eventFilter(QObject *object, QEvent *event){
    if( event->type() == QEvent::MouseMove){
        QPushButton *btn = qobject_cast<QPushButton*>(object);
        btn->setStyleSheet("background-color: black;");
    }
    if( event->type() == QEvent::MouseButtonPress){
        QPushButton *btn = qobject_cast<QPushButton*>(object);
        btn->setStyleSheet("background-color: white;");
    }
    return QMainWindow::eventFilter(object, event);
}

btns1[i][j]->setMouseTracking(true);

QPushButton:pressed {
     background-color: white;
 }
QPushButton:hover {
     background-color: black;
 }