Do I need to record employee wages payments bank transactions to Quickbooks, after I finalized pay run?

Thanks for posting to the Community, @cocazxl.

No, after you finalized a Pay Run in QuickBooks, journal entries are exported to QuickBooks containing the Pay Run data. It will automatically record payments to QuickBooks after you finalized the Pay Run.

You can check this article for more information: Understanding W1 and W2 tax codes in QuickBooks Payroll. You can check there the allocation of the codes to the following transaction types.

Please know you can always leave a message to this post if you have additional questions. I'll be happy to help you. Take care!