You talk about "Alertmanager payload" and "the integration" without saying what this integration is (a webhook? something else?).
The string "incident_key" does not appear anywhere in Prometheus source code. It does appear in Alertmanager, but only in the context of Pagerduty. You didn't mention Pagerduty at all in your post.
If your issue *is* with alertmanager->pagerduty integration (which I'm just guessing), then the documentation for what you can configure is here:
and the logic for how incident_key is generated is in the source:
It is hard-coded to key.Hash(), where key is an instance of notify.Key, which comes from notify.ExtractGroupKey
If you want to customise this, then I suggest you write your own webhook receiver, which accepts the (fixed) JSON
webhook format, and then your code generates a new payload towards some other system, in whatever format you like.