If you charge a client's credit card outside of Full Slate, you can still mark the appointment as paid through Checkout. Just select "Record Credit Card" from the drop-down of options during Checkout.
Full Slate is integrated with Stripe, PayPal, Authorize.net, and PayPal Payflow. For more on this, please click here.
As described here, Full Slate is not currently integrated with QuickBooks Payments, but you can sync payment details to QuickBooks Online and process the payment using QuickBooks Payments there.