You can trigger email surveys, SMS surveys or even in-app web or mobile surveys from Intercom using webhooks. Here is a video with steps on how to configure the two services.
In addition to this webhook integration, Wootric also offers a Messenger App that delivers surveys in Intercom Messenger itself. And, through our bi-directional integration, survey responses can appear in Intercom user records. Please see the details at https://www.wootric.com/integrations/intercom/