You may encounter an error when trying to delete a note from conversation via an API call (i.e., .../api/v1/cases/<id>/notes/<id>):
Message: No matching resources were found
This issue may happen when the note that is displayed in the conversation is linked to an organization.
To identify if this is the case, call a GET API call for the required conversation and check the resource URL of the required note - as you can see on the screenshot below, the resource URL for the note is pointing to Organization note; you should use the Resource URL value to delete the note.
For more information, refer to the article Deleting an Organization Note via API.