Intuit does offer something called (I think...) "QuickForms" that does payroll forms.
It is separate from the PS income tax software.
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Looks like QEF only does 1099s and W2s, I don't see the quarterly reports shown on this info page.
If you are using a QB program to manage the accounting data, then you can pay for a Subscription for Payroll, which enables tax tables and the payroll tools, in that FEIN file, including the ability to e-pay and e-file those forms. I recommend asking Intuit's Payroll sales. The level for desktop that includes the e-pay and e-file is called Enhanced payroll.
"Level Up" is a gaming function, not a real life function.