The register entries are easy to make, but that's not your only issue so I will address both.
To make your register entries for the check, you can make a journal entry. Debit payroll expense for the gross pay and debit payroll taxes for the employer's share of payroll taxes. Credit cash for the net pay and credit payroll tax liabilities for the withholdings and employer payroll taxes. This will allow you to reconcile and properly record the payroll check in your books. In order to ascertain the amounts, you should refer to the payroll summary report for the paycheck in question (Reports -> payroll -> payroll summary; filter the date for the specific paycheck). If you're not familiar with how to make journal entries in Quickbooks or which accounts you are using to track payroll, then this may require a little more hands on attention so I've provided some additional detail below. PS - Journal entries do not impact Quickbooks internal payroll tax calculations so a warning message will appear, but since you are using their external payroll service this is not a big deal in and of itself.
The problem comes with Intuit's payroll tax calculations. It's unclear whether or not your tax returns were properly filed. If not, amended returns may be required for state filings (federal filings for July payrolls generally are not due until October so no amendment would be needed, but Intuit may not have not paid in enough for July's payroll so an additional tax withdrawal may be necessary). To get to the bottom of this issue, you will need to contact Intuit's payroll department being that they are filing your returns. I believe they guarantee the calculations and filings, so they should straighten this out at no cost to you.
If Intuit properly prepared your tax filings and made the necessary tax payments, then the only issue is making the journal entry. The following article will be helpful: https://community.intuit.com/questions/1145322-recording-payroll-transactions-manually
Taken together, this should solve your issue. Best of luck!