Intercom's API lives at https://api.intercom.io (api.eu.intercom.io and api.au.intercom.io for other regions). An Intercom-Version header pins a version; without it your app's default applies.
Get an access token from your app in the Developer Hub and send it as Authorization: Bearer <token>. In the examples it's written {{INTERCOM_TOKEN}}.
Full reference: https://developers.intercom.com/docs/references/rest-api/api.intercom.io
curl https://api.intercom.io/me \
-H "Authorization: Bearer {{INTERCOM_TOKEN}}" \
-H "Accept: application/json"
curl "https://api.intercom.io/contacts?per_page=10" \
-H "Authorization: Bearer {{INTERCOM_TOKEN}}" \
-H "Accept: application/json"
curl https://api.intercom.io/conversations \
-H "Authorization: Bearer {{INTERCOM_TOKEN}}" \
-H "Accept: application/json"