In my compilation, I only have two names for the letter "Q".
The first name is "Qualis/Qualls, Quarrels, & Quarles" and the second name is "Quinn".
The Quinn entry is for Rev. William T. Quinn and I am not sure if he was Caucasian or Black. However, he officiated at the 1874 marriage of James Nickell and Dolly/Molly Thompson (in Monroe County) and the 1874 marriage of Robert Carter and Jennie Jackson (also in Monroe County). The only likely William Quinn that I could find was a Caucasian male who was born in 1844 and who lived in Forest Hill, Summers County. He was a farmer who appeared in the 1880 census.
As for the "Qualls" group:
The "Qualls" spelling was used for a 1904 marriage and might be the correct spelling for this family.
Some entries are for -
Alice Cosby Quarles, the daughter of George and Nancy Cosby. She can be found on the 1920 Census at Avistown, Summers, at age 66, widowed, and the grandmother of Martin French (22) and on the 1930 Census at age 65, in Hinton, Summers County, with her son Alexander (45). She died on 11/12/1932 at Summers. In 1904 she appeared as the mother of Rosa B. Qualls who married George W. Scott. She can also be found on West Virginia Death Certificate # 1200 as the mother of Rosa Scott.
Doc/Lowry Qualls was the son of Matt & Polina Qualls. He appears on the 1900 census in Fayette at age 7 and is enumerated as Lowry. He died on 1/9/1939 in Pocahontas.
There is another "Doctor Quarles" who appears on the 1880 Census in Greenbrier at age 60, b. about 1820 in Va.
For more information on this family see BLACK RESIDENTS OF GREENBRIER, MONROE, POCAHONTAS, AND SUMMERS COUNTIES by Carol L. Haynes or contact me at firstname.lastname@example.org.