This is a very well-known sign of stress/anxiety in Quaker Parrots, and if you do a Google search you'll see how common it is. My Quaker, Lita, will do this only when she does something she knows she shouldn't, usually it's when I reach into her cage to change her water or food and she nips at my hands (they are extremely territorial birds as well, more so than most other species). As soon as she nips my hand, before I can even say "No Bites!" to her, she immediately lifts her foot to her beak and starts chewing on her toenails. Once the moment has passed and her anxiety has gone, she stops. That's the only time she does it.
Once I noticed that Lita was doing this only during times of anxiousness/stress, I did some research online, and within minutes I discovered that this behavior is extremely common to Quaker Parrots when under stress. Quakers who have been re-homed often spend all day long chewing on their toenails/feet during their first few days/weeks or even months of being in their new home, and as they settle-in and adjust the behavior gradually stops over time. So it's likely that this is the reason that your Quaker is doing this, for some reason they are feeling anxious or stressed about something.
How long have you had your Quaker?