Do any of the 3,000+ county names in the U.S. begin with the letter Q?

Yes indeedy. There are four of them altogether. Quay, Queen Ann's and two Quitmans.
http://www.happychild.org.uk/usacounties/alpha/Q.htm

There is also Queens County in New York State. Queen's is also a borough of New York City. It is the largest borough by area and is the 2d most populous county in New York State.



https://en.wikipedia.org/wiki/Queens

The link from my answer above lists all of the counties of New York, but does not mention Queens. However, a little digging shows that Queen's is mentioned as "a borough of New York City, coextensive with Queen's County in south western New York" (Several references say the same thing).Therefore, the omission of Queen's from the first data base is curious and thank you to MiraJane for bringing it to our attention.

Here is a link to another database. This one contains Queens Co.

https://public.opendatasoft.com/explore/dataset/georef-united-states-of-america-county/table/?disjunctive.ste_code&disjunctive.ste_name&disjunctive.coty_code&disjunctive.coty_name&sort=year