Start two different WebDrivers from your test. Start a FirefoxDriver (or ChromeDriver, or whatever driver you want to automate the web browser porition), and start your second driver as some AppiumDriver (e.g. AndroidDriver).
Use your desktop driver to navigate to your website and enter the needed details to generate the QR code. Then, have your AppiumDriver start the QR reader application on your device. Use Appium's tools to wait for when the QR code has been read in and your app is ready to continue.
This should give you the basic idea. There's no rule saying you can't use two different drivers at the same time. Just make sure your phone has a decent camera to pick up the QR code.
Of course, this does seem like a rather unstable setup for testing. Perhaps there might be a tighter way to send QR codes to your phone.