The first three Giants-Yankees matchups were in 1921, 1922, and 1923. However, in 1921 an 1922, the teams played in the same stadium at the Polo Grounds. The series was nicknamed the "Battle of Coogan's Bluff." The 1923 matchup was called an "all-New York World Series."
"Subway Series" was used at the 1934 all-star game that featured many Giant and Yankee players. However, the New York teams did not both make it to the World Series that year. The first "Subway Series" was actually the fourth, in 1936.
The subway fare used to be a nickel, and the term "Nickel Series" appears earlier than "Subway Series."
4 August 1927, Atlanta Constitution, pg. A3:
Most New Yorkers would rather have all the games of the 1927 world series within a nickel's ride of their homes.
25 September 1928, Bee (Danville, Virginia), pg. 3, col. 3:
I feel that I am violating no confidence in saying that it looks like New York in the American League. The other half of the proposed subway series may or may not be closer to its goal line by tonight but no one can say that their chances are only nominal.