Usually, "Add to cart" button on Quick View 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 > Customize > Extras. Then it'll be possible to update the cart count of your theme when customer clicks on "Add to cart" on Quick view popup.
Cart Count is the number showing up above cart icon to show the number of items in cart.
Watch this tutorial video or Follow the steps below to find the CSS selector of your cart:
Step 1: Make the cart count appear
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 count number will appear.
Step 2: Find CSS selector of the Cart Count
Right click on THE NUMBER of the cart count → Inspect.
The developer tool will appear and auto detect the HTML tags of your cart 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 count.
Step 3: Paste to Quick View
Head back to Quick View → Customize → Extras → paste to CSS Selector of your cart
Save. Done! You've successfully integrate Quick View with your default Cart.