Mark,
According to an article in the 1934 issue of the New York Sun, on May 31st "The Quogue Field Club opened its 47th season yesterday..." Meaning that it was founded by and opened in 1887.
Golf was played at the club as early as 1897. The 7/28/1897 issue of the NY telegram reported: "Quogue's season has reached its height... Those interested in the weekly golf tournaments have turned their attention to the tennis tournament. However, golf seems to be the chief outdoor amusement at Quogue..."
Their first professional MAY have been James Floyd Sr. who left that position in 1905.
The original course was 9 holes and approximately 2,500 yards in length. After 1923 it was expanded to 18 holes and when finished by 1925 it measured 6,200 yards and would remain the same well into the 1930s at least...
There are a tremendous amount of old newspaper articles that one can cull through about Quogue but most of them deal with societal functions. For example, it appears that for decades after its opening a highlight of their season were the "Quogue Quips," evidently a variety show put on at the club...