I have located only one museum in New Zealand with an airship exhibit. If I encounter other museums in New Zealand which are verified to have an airship exhibit they will be included here.
At the Auckland War Memorial Museum there is a small display about airships consisting of a few displays and small artifacts related to the WW I bombings in England.
Photo credit: Joyofmuseums - Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=63033412
Photo credit: Greg Hatswell
Above are two photos of the Zeppelin displays at the Auckland War Memorial Museum courtesy of Greg Hatswell. The displays describe WW I Zeppelin air raids on London and include bomb fragments, a zeppelin bomb-release mechanism, a cutaway of the bomb type dropped from a Zeppelin, a small section of a duralumin girder from a downed zeppelin, and zeppelin telgraph component used to pass messages from the zeppelin bridge to an engine room.
Website: Auckland War Memorial Museum
The museum is located at (Lat Lon) -36.861401, 174.777820, address The Auckland Domain, Parnell, Auckland 1010, New Zealand.
The location of the location of the museum in Google Maps: