Usually, "Add to cart" button on "Quick view & Quick shop" is separate with your theme's cart.
To integrate with your cart, you need to find the CSS selector of your cart and paste to Quick view & Quick shop → Content → Options → Cart item count ID, Cart total price ID. Then it'll be possible to update the cart item count or cart total price of your theme when customer clicks on "Add to cart" on Quick view popup.
When there's no item in cart, your cart can show "0" number or show nothing.
First, you need to go to a product page → add product to cart → the cart item count number will appear.
Right click on THE NUMBER of the cart item count → Inspect.
The developer tool will appear and auto detect the HTML tags of your cart item count (usually starting with <span>).
Right click on the <span> (or what highlighted) → Copy → Copy selector.
You've successfully copied the CSS selector of your cart item count.
Right click on THE PRICE of the cart total price → Inspect.
The developer tool will appear and auto detect the HTML tags of your cart total price (usually starting with <span>).
Right click on the <span> (or what highlighted) → Copy → Copy selector.
You've successfully copied the CSS selector of your cart total price.
Go to Quick view & Quick shop → Content → Options → Cart item count ID, Cart total price ID → paste to CSS Selector of your cart.
Click Save.
Done! You've successfully integrate Quick View with your default Cart.