Yes, you can send a Quick Fax from a Case. A Quick Fax sent from a case will have the barcode that was generated when the case was originally created. This does not apply to Quick Faxes sent from other areas of the web application.
To send a Quick Fax from a Case, open the Case panel and select the Quick Fax icon.
Complete the required fields and and select Send Fax. This will log the transmission as an activity in the Case Activity History.