customGraphicsScene *customScene;

ui->graphicsView_2->setScene(customScene);  

Think about it! customScene is just a pointer to some customGraphicsScene (and an uninitialized pointer at that). You have not allocated any actual customGraphicsScene instance for it to point it. I did say earlier:

create a new QGraphicsScene and assign it to the QGraphcisView via graphicsView->setScene().

you need to:

customGraphicsScene *customScene = new customGraphicsScene(this);
ui->graphicsView_2->setScene(customScene);

But a couple of points:

• Your customGraphicsScene *customScene is a local variable in the MainWindow constructor. Once that exits the local variable no longer exists, you have allocated a customGraphicsScene but you really can no longer get at it to do anything! You will want that pointer to be a member variable in your MainWindow class.

• class customGraphicsScene : public QGraphicsScene: make all your classes start with an uppercase letter. All Qt's do, and this is standard practice for C++. Your variables, such as [now] CustomGraphicsScene *customScene, should indeed start with a lowercase letter.

class customGraphicsScene : public QGraphicsScene {

    Q_OBJECT

protected:

    void mousePressEvent(QGraphicsSceneMouseEvent *event) override;

};

class source file:

void customGraphicsScene::mousePressEvent(QGraphicsSceneMouseEvent *event) {
    if (event->button() == Qt::LeftButton) {
        qDebug() << "clicked!";
    }
}

mainwindow implementation:

MainWindow::MainWindow(QWidget *parent)
    : QMainWindow(parent)
    , ui(new Ui::MainWindow)
{
    ui->setupUi(this);

    customGraphicsScene *customScene;

    ui->graphicsView_2->setScene(customScene);   
}

MainWindow::~MainWindow()
{
    delete ui;
}

• To access the mouse you will need to use mousePressEvent() and mouseMoveEvent() methods. These are available on both the QGraphicsView and QGraphicsScene. A mouse event is sent first to the view and then to its scene. In both cases these are virtual protected methods. That means you have to subclass the view/scene to intercept them.

• If you wanted to use them at the view level you would need to create your own QGraphicsView subclass. Since you want to use Designer you would need to use its promotion feature to create a subclassed QGraphicsView at design-time.

• However, I think you will be happy using them at the QGraphicsScene level, where they have been translated into a QGraphicsSceneMouseEvent. So you will need to create your own subclass of QGraphicsScene to override the methods. Since you assign the scene to the view in your own code it's easy to just assign your subclassed instance. In those overridden methods you do whatever to the points of your curve.

• If you need to see mouse events on a QGraphicsItem, like @SGaist mentioned, then similarly there are the same methods on those and you would subclass the items you add to the scene.

So to repeat: I think you will want to use mouse events on the graphics items or scene rather than on the view.

P.S.
Just came across how to use mouse move event for QGraphicsScene? from years ago. Might be worth looking at.

am having trouble figuring out how to get started on implementing this into an existing mainwindow UI.

• In Designer from the toolbox at the left, section Display Widgets, drag a Graphics View onto the central widget in your MainWindow. This is a widget, just like all the other widgets there (e.g. Label, Line Edit, Frame etc.), so you can put it wherever you could put those and do all the usual layout stuff and so on. This gives you a QGraphicsView widget named graphicsView.

• A QGraphicsView needs a QGraphicsScene as its "model", but you cannot create that from Designer. Somewhere in your code --- probably in the MainWindow constructor after the ui->setupUi(this) --- create a new QGraphicsScene and assign it to the QGraphcisView via graphicsView->setScene().

• Now you have a graphics view with a graphics scene behind it. You can start creating your QGraphicsItems and adding them onto the scene. They will be visualized in the view. You can do whatever to the scene or view to alter its look and behaviour.

One way you can do that is using the Qt Graphics View framework.
Create a custom QGraphicsItem that you will use to move the curve's reference points and a QPainterPath based on them.