What I would suggest is, attach your iPad to your computer, to a full local backup and then, having disabled Find my iPad on the iPad, use the Restore to factory default via your computer (not via the iPad). Once that is done, your computer will have downloaded and installed the latest iPad OS versus available and compatible to your iPad Air 2.
Now, setup your newly refreshed iPad as if it were a new one for you, using your Apple ID and see if you can use the new Quick Notes. If you can, it was likely a stuck setting or issue from your old setup. No need to worry, you can still overlay your backup from above overtop the newly installed OS and be back to a fully as was setup iPad (with the fixes though now in play) of yours, likely within an hour.
I hope this helps.