I subclassed a QPushButton so that I was able to re-implement the paintEvent(QPaintEvent *paint) method also advised by the official documentation.
Below is the sequence of operations:
a) After I launch the application the button is below:

b) This is after I hover on it:

c) then I click the button:

d) and finally I release the mouse

e) go away from the button

However the problem is that the QPainterPath with which I designed the green box it should be red after I release the button. And, of course, it should become green again after I click again the button.
Below the code:
custombutton.h
class CustomButton : public QPushButton
{
    Q_OBJECT
public:
    CustomButton(QWidget *parent = nullptr);
    ~CustomButton();
    QString FirstName = "MACHINE";
    QString LastName = "CONTROL";
protected:
    void paintEvent(QPaintEvent *);
};
custombutton.cpp
CustomButton::CustomButton(QWidget *parent) : QPushButton(parent)
{
    setGeometry(150, 150, 110, 110);
    setAttribute(Qt::WA_TranslucentBackground);
    setStyleSheet(
        "QPushButton{background-color: lightGray;border: 1px solid black; border-radius: 5px;}"
        "QPushButton:hover{background-color: gray;}"
        "QPushButton:pressed{background-color: lightGray;}");
}
CustomButton::~CustomButton()
{
}    
void CustomButton::paintEvent(QPaintEvent *paint)
{
    QPushButton::paintEvent(paint);
    QPainter p(this);
    p.setRenderHint(QPainter::Antialiasing);
    QPainterPath path;
    path.addRoundedRect(QRectF(15, 15, 80, 5), 2, 2);
    QPen pen(Qt::black, 0.3);
    p.setPen(pen);
    p.fillPath(path, Qt::darkGreen);
    p.drawPath(path);
    p.save();
    p.drawText(QPoint(20, 60), FirstName);
    p.drawText(QPoint(20, 80), LastName);
    p.setFont(QFont("Arial", 10));
    p.restore();
}
mainwindow.h
class MainWindow : public QMainWindow
{
    Q_OBJECT
public:
    MainWindow(QWidget *parent = nullptr);
    ~MainWindow();
public slots:
    void onClickedButton();
private:
    Ui::MainWindow *ui;
    CustomButton *newBtn;
};
mainwindow.cpp
MainWindow::MainWindow(QWidget *parent)
    : QMainWindow(parent)
    , ui(new Ui::MainWindow)
{
    ui->setupUi(this);
    newBtn = new CustomButton(this);
    newBtn->show();
    connect(newBtn, &QPushButton::clicked, this, &MainWindow::onClickedButton);
}
MainWindow::~MainWindow()
{
    delete ui;
}
void MainWindow::onClickedButton()
{
    QPushButton* target = qobject_cast<QPushButton*>(sender());
    if (target != nullptr)
    {
        QPainter p;
        p.setRenderHint(QPainter::Antialiasing);
        QPainterPath path;
        path.addRoundedRect(QRectF(15, 15, 80, 5), 2, 2);
        QPen pen(Qt::black, 0.3);
        p.setPen(pen);
        p.fillPath(path, Qt::darkRed);
        p.drawPath(path);
        p.save();
        p.restore();
    }
}
As you see in the MainWindow I created a function onClickedButton() which I thought would have triggered the QPainterPath into red after clicked. And in order to do that I casted it into an object (qobject_cast). But unfortunately it didn't work as expexted.
                        
The solution is to create a flag that indicates the color and call update() for repainting: