You can send out your own questionnaire or a link to a survey using Intercom's Auto messaging feature. The auto message aspect of this will allow you to send your survey to a targeted group of users as soon as they fall into your specific research group. You should define this group by using the filters. You will need to link to a third party survey however because currently, there is no survey specific survey tool in Intercom.
This is something that we get asked for quite regularly and we're considering whether we should build a specific survey tool. For now though, if it's something you feel you need, please let us know by mailing firstname.lastname@example.org We're collecting requests for it.
While we don't offer customizable surveys we do have the option of reactions in the Intercom Messenger. Reactions collect valuable feedback to your messages from users without requiring you to respond in the same way text feedback does.