Facebook filters out messages sent to an Instagram tester account from users who don’t have a role on the app. To see conversations via the Conversations API, the messages must come from an Instagram account linked to a Facebook account with a role (e.g., Admin, Developer, Tester) on the app.

Just add your Instagram account as a tester in the Facebook app, and you'll start seeing the messages in the API.

See: Instagram Messaging API – Rollout