You certainly could simply record deposits into QB as general income but i fear you will be missing half of the transaction and if your billing system is only billing then ddfinitrly you lose any automation of expense.
Other than just posting deposits as income your possibilities would be
1. Switch to billing from within quickbooks with added capability of customers being able to pay directly from email
2. Switch to a billing system that syncs to quickbooks so that you only enter data once.
3. Switch to a billing system that includes its own accounting and unsubscribe to quickbooks
