You can do what you want to do with a webhook (new feature in September's release). Webhooks are set up like emails or notifications. You can specify an action to take (including adding a record) when a field is changed on a PM (set to completed). If you go to settings in PM, then click on webhooks you will see a yellow banner that has links for further information.
I've attached a partial sample to get you going..
If you're unfamiliar with the QB API calls you may need to get some assistance from someone who is. Of course we'd be glad to help.