How to query conversation by type on the api

You should be able to accomplish this using the Search Conversations endpoint. 

Your query would need to search by “recipient”, using the channel address (not channel ID)  (see the conversation search syntax). 

As an example, we can call GET /channels to list all our channels via the API. From there, find your Intercom channel, and look at the address field. For the sake of this example, let’s imagine the address value is abc123. 

Our API call to search for conversations sent/received via that channel would then be the following;

GET https://api2.frontapp.com/conversations/search/recipient:abc123

I hope that’s helpful!