There’s no Avenue Q in Brooklyn, and there’s no E or G, either; E became Foster Avenue and G became Glenwood Road long ago. The reasons for those switcheroos are murky, but the reason there’s no Avenue Q is quite clear.
In the late 19th Century, the street system of the towns of Flatbush, Gravesend and Flatlands was crystallized: east-west avenues would carry letters of the alphabet, but unlike Washington, DC’s lettering, these would be Avenues, not Streets. North-south streets would be numbered East and West, with West streets located west of Dahill Road/Gravesend, later McDonald Avenue, and East streets on the other side.
Avenue Q, therefore, took its place between Avenues P and R. A tragedy befalling the former First Family of the United States, however, initiated a change. Quentin Roosevelt, one of ex-President Theodore Roosevelt’s six children and his youngest, was killed at age 20 during aerial combat over France on Bastille Day, July 14, 1918.
As a tribute to Quentin Roosevelt, Avenue Q was renamed for him sometime after Armistice Day, 11/11/18. In Queens, a new avenue built below the new Flushing El had been named Roosevelt Avenue in 1914 for Theodore Roosevelt.
I’m not sure whether authors Robert Lopez and Jeff Marx were aware of Avenue Q’s absence in Brooklyn when the wrote the Broadway musical Avenue Q, but I suspect they may have.