Your program can do anything you code it to do if you connect a slot to the button's clicked() signal. And using a button click to start a simulated deal of hands of cards is only one way you might do it. The two are related only if you chose to use a button for that purpose. Don't get locked into confusing how you accomplish a goal with the goal itself.

As for displaying the game cards, the Qt Graphics / View framework is one way to do that. You can use QGraphicsPixmapItem instances with JPG or other images to represent the cards. To simulate dealing, you can animate the position of the cards using the Qt Animation framework.

I would suggest that you start simple. Instead of jumping into the deep end, develop a simulation that uses simple rectangles instead of card images. Learn how to use the Graphics / View architecture to create a QGraphicsScene containing the rectangles and to place the rectangles where you want them in the scene (and scaled to fit on the QGraphicsView). When you get that working, add code to move the rectangles around using a controlled animation. When that is working, find playing card images and replace the rectangles with pixmap items and those images. When that is working, add the code that implements the rules of your card game and the user interface elements your users will need to control the game and display the scores, etc.