what significance of the tune coming from the piano?

Fitzgerald introduces the melodies played on the piano at a crucial point in "Winter Dreams" when Dexter encounters Judy Jones as a mature woman for the first time. The music, significantly, is from past summers and from the years when Dexter was still immature himself but was formulating his life's goals.
The music symbolizes both the innocence of youth and the boundless promise the young believe the world holds for them. In "Winter Dreams," Judy is another version of the dream-girl figure who appears again and again in Fitzgerald's fiction. Usually it ends up that disillusionment sets in for the protagonist, either with the girl herself or the overall situation. The most famous example of Fitzgerald's dream-girl type is, of course, Daisy in The Great Gatsby.
But although Dexter's enchantment with Judy is just getting into full swing (and on the golf course!), the music is also a subtle foreshadowing that his attachment to her will not last. It represents a "brightness and a glamour he might never know again." In other words, it's the climax of the story. From this point the reader senses that the promise will remain unfulfilled. Fitzgerald's implication is that the reverie of youth, as wonderful as it is while it's happening, is over almost as soon as it begins.