"but now what I would *like* to do, is convert this employee's pay data into the Quickbooks Payroll."
You cannot "move" the vendor name. Put an X in the beginning and then make it Inactive, so it doesn't show for selection.
You cannot Move anything to payroll, either. And, you don't have to. You have accounting data; there is nothing to Move about it.
Don't delete and re-enter. You already have Activity. Unless you really have nothing else to do this month?
Set up the Employee and your YTD existing info is going to be used to Populate payroll in the Payroll setup wizard; pay attention to the Options button NOT to affect accounts. You want everything to get its "background" for YTD, but NOT also hit expense, bank and liability again.
Please see my screenshot attached.