No, they did not.
Actually yes they did. The deed to Fenway Park was part of the deal when the Red Sox sold Babe Ruth to the Yankees
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It came on August 31, 2011 off of Luis Ayala of the New York Yankees in the bottom of the 8th inning.
The first no-hitter in Yankee history was pitched on April 24, 1917 by lefthander George Mogridge against the Red Sox at Fenway Park. He gave up one run. The Yankees won the game 2-1
Joe Lucey debuted on July 6, 1920, playing for the New York Yankees at Polo Grounds IV; he played his final game on June 25, 1925, playing for the Boston Red Sox at Fenway Park I.
Buddy Rosar debuted on April 29, 1939, playing for the New York Yankees at Yankee Stadium I; he played his final game on September 19, 1951, playing for the Boston Red Sox at Fenway Park II.