The easiest way to create a service and integration is in the PagerDuty web UI, however, you can create a service with a vendor-specific integration via the API.
You’ll need to know the vendor_id. Although we haven’t documented the vendor API, it follows the same pattern as our other APIs. Here's an example using PDJS to look up vendor_id: http://jsfiddle.net/8fxudqbj/6/.
Note: You will need a v1 API key to use this JSFiddle.
Here is a list of common vendor_ids:
After finding the right vendor_id, you'll need to POST the new service to the services endpoint with type: "integration", vendor_id: "PQFCJER"