The New Jersey tax return has no provision for a third party designee. You can't enter one because there is no place for it on the New Jersey tax return. The New Jersey tax return in TurboTax doesn't show a third party designee because the state tax return form does not have a place for it.
The check box that you are looking at says "I authorize the Division of Taxation to discuss my return and enclosures with my preparer." That means a paid preparer who prepares and signs the return, someone you paid to prepare your tax return. If you are filing your tax return yourself with TurboTax you do not have a preparer. So there is no way to check that box in TurboTax because it never applies to a TurboTax return. TurboTax cannot be used by paid preparers.