When syncing payments to QuickBooks Online from WooCommerce, we support syncing through a variety of different gateways, but we do not do any sort of Payment Processing.
However, this is possible with a separate plugin/gateway. Our sync doesn't handle payment processing, but this separate gateway allows you to use your Intuit merchant account to process these payments.
Note: There are a few different plugins/gateways out there, but the most popular one (Intuit Payment Gateway) is linked here.
This extension supports merchants using an Intuit Payments account for eCommerce processing.