As omotenashi and Dave in Saitama said, you need to go to the company that issued the card, and if you have the green/silver color card type that is shown at the top of the page (link provided by you), or the NEX & Suica package one you buy at Narita airport, then that means JR East.Unfortunately, Haneda Airport station is not a JR station, so you cannot get a refund for it there. So, you need to go to one of the counters of JR East station "before" you get to Haneda Airport.
I don't know exactly where you are starting from to get to Haneda airport, but if you take the monorail from Hamamatsucho station, Hamamatsucho station would be the last JR East station you change trains at, and if you take the Keikyu line, Shinagawa station *may* be the last JR East station you change trains before you get to Haneda airport.
So for the last train ride to Haneda airport, you need to buy a single ticket to take either the monorail or Keikyu line.