There is no QuallS reservation in NC. What is there is QuallA Boundary, which afaik, is not related to the Qualls name. My understanding/memory is that there was a trading post operated by a woman named Polly or Paula, which was hard for the Cherokee to pronounce and their pronunciation was Qualla.
That said, the Qualls/Quarles names occur on various Cherokee rolls, both for the Eastern Band in NC and the Western band in Oklahoma.
There was a 'village' known as Qualls Oklahoma, about 15 miles south of Tahlequah, capital of the Western Band of Cherokee. There was a Qualls post office established in 1909. It was perhaps more properly White Oak but that name was taken so the post office was named Qualls after the first postmaster.