No, if you are referring to any country club it is NOT capitalized. I went golfing at the country club. If it is a specific country club, you capitalize it. I went golfing at the Summertime Country Club.
Marlboro Country Music club
Newport Country Club was created in 1893.
Visalia Country Club was created in 1922.
Kloof Country Club was created in 1927.
Westchester Country Club was created in 1922.
Muthaiga Country Club was created in 1913.
Lakewood Country Club was created in 1912.
Country Club Plaza was created in 1923.
Haworth Country Club was created in 1965.
Ridgewood Country Club was created in 1890.
Woodcrest Country Club was created in 1929.