You can listen to jazz in Budapest in several places, and in various genres, mixed with electronica, traditional jazz, world music inspired jazz and so on.
The Budapest Jazz Club is quite popular, as is the Palace of Arts in Budapest with all sorts of jazz events all year round. But keep in mind the really cool A38 Music Ship, an unusual venue for concerts: a standing ship at the Petofi Bridge on the river Danube in Budapest. Lonely Planet hot tip to go to A38 Budapest as much as you can, provided you love music, and a superb atmosphere.
Also Budapest Music Center is a recent addition of Budapest concert halls. High quality, great location.
NPM, the Norwegian talent is giving a jazz concert on A38 in May as part of his Eastern European Tour.