Hi,
On the product page, there is a + and - sign to change the value in the quantity input field. That doesn't require any modifications.
For the quantity of the products in the cart module, you will have to edit the file "cart" of the view "product" via the menu Display->Views and add you AJAX code there. You can just call the add to cart in AJAX so that it returns the new content of the HikaShop cart and replace the module with that new content.
You can look at the file adminsitrator/components/com_hikashop/helpers/cart.php to see how we did it for the product page add to cart button.