At this time, you can get paid (A/R) with Melio and sync to QuickBooks Desktop, and the other way around.
But you cannot sync bills from QBD into Melio and pay them (A/P) with Melio and/or push paid bills back to QuickBooks Desktop.
Your Accounts Payable (e.g. bills) and Accounts Receivables (e.g your invoices) both sync between Melio and QuickBooks Online as well as Melio with QuickBooks Mobile.