I did this back in 2014 for our incidents. I added a button on the Incident form that executed an outbound REST or SOAP action I can't remember into the ServiceNow Table API endpoint and POST https://instancename.service-now.com/api/now/table/incident.
We would send the existing remedy ticket number into a custom field (might use correlation id now) and update the Remedy ticket with the ServiceNow INC number returned in the response. Then close the remedy incident with a filter and send the user who created the incident a notification with the new ServiceNow number.
