getMusicPlaying is set to false whenever someone touches the button, but libgdx runs show method only once, at the start of the game, hence the music will keep playing forever.

I tried to use the render and batch.begin instead of show() same result.

private boolean musicPlaying = true;

button.addListener(new InputListener() {
public boolean touchDown(InputEvent event, float x, float y, int pointer, int button) {
 return true;
}

public void touchUp(InputEvent event, float x, float y, int pointer, int button) {
 setMusicplaying(!musicplaying);
}
});

}
});
}

public void show() {
 if (getMusicPlaying ==true)
  music.play();
}