Your payroll subscription is linked to your company file.  After installing the new QB version and downloading the latest updates, restore your backup from the old version, then download payroll updates.

http://payroll.intuit.com/support/kb/1001166.html?rSearchTerm=payrollupdates

Resolution for 'ax subscription issue: 'I converted from QB Pro 2016 to QB Premier 2016 (Contractor’s Edition).. .how do I move my tpayroll ax subscription?' available: Yes.

The ax subscription issue or error code is a known issue in Quickbooks Online (QBO) and/or Quickbooks. Support for this issue is available either by self-service or paid support options. Experts are available to resolve your Quickbooks issue to ensure minimal downtime and continue running your business. First try to resolve the ax subscription issue yourself by looking for a resolution described in the article. If it is a complex ax subscription issue or you are unable to solve the issue, you may contact us by clicking here or by using other ax subscription support options.

For more information about resolving issues related to ax subscription in Quickbooks/QBO or to get support for the issue 'I converted from QB Pro 2016 to QB Premier 2016 (Contractor’s Edition).. .how do I move my tpayroll ax subscription?', click the link below: