Create a private note when creating an incident from AAQ page

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I have an AAQ page for submitting requests (create an incident) where I populate some information in the incident.thread field. I am wondering if there is a way to also add a private note to the incident on creation as well?

I will be capturing some basic information through the browser like OS, browser version, etc. The idea is to have it in a private note so that it's not included with the customer response which they will see (looks messy) when they view their request through the portal. Therefore the information to be added to the incident via private note directly from the submission of the AAQ page.