Can I set quick books payroll to automatically pay the weekly taxes due to the federal and state governments?

Hi benher58, 

Glad you visit the Community today. QuickBooks Payroll Enhanced will only calculate payroll taxes for you and give you the option to pay them through QuickBooks. 

You'll want to enroll in our electronic services in QuickBooks Online so you can pay or file your federal and state payroll taxes online.

While the Full-Service Payroll, QuickBooks automatically submit the payment for you. I'd recommend contacting Full-Service Payroll (IFSP) customer support to check your payroll taxes and payments.

For future reference, you can check out this article: Manage automatic tax payments and form filings. You can explore your options in paying and filing your payroll taxes. 

I'm always around to help you more with your additional concerns. Take care!