Hmm. At a glance, implementing exactly what you’re looking for isn’t straightforward—and it might cost you a few subscription fees.
Zapier can do actions based on FB Messenger messages:
You could use that to send a trigger to a Pushcut server…
…which could then create and append to Notes notes.
Trick is, you’ll probably need a Zapier subscription, a Pushcut subscription, and a dedicated iOS device on which to run the Pushcut server.
You could cut the Pushcut part by using Zapier to push messages to something like Google Sheets, and then manually run a Shortcut (you can set up regular reminders to initiate this) to detect new additions to the spreadsheet and act on Notes.
And that would still require, I think, a Zapier subscription.
This is just off the top of my head. Maybe there’s a simpler way I’m not thinking of.