There are MANY species of fish from the Great Barrier Reef in captivity and I won't even attempt to make a list because it would be very, very long. Some of these are widespread species that originate elsewhere (e.g. the Solomons); some originate from Australia itself. There are companies that catch and export marine fish with some regularity there and rarely even Australian cold-water marine species are exported. The Australian laws and restrictions on this animal group are less tough than the similar laws/restrictions for Australian birds, reptiles and mammals.
Relatively few freshwater fish from Australia are kept elsewhere; the majority are rainbowfish. There are dedicated aquarium groups for this Australian-New Guinea fish family and many are now captive bred. Most other Australian freshwater fish fauna is rather dull looking and there isn't much interest in them. The Australian lungfish is strongly protected. It's very long-lived and most in aquaria outside Australia arrived decades ago, but exports of captive bred is still possible with the proper permits. Other freshwater fish of Australian origin that I know are in captivity elsewhere today are Mogurnda gudgeons, galaxias, Chlamydogobius gobies, spotted bonytongue (bred in aquaculture), and Murray cod (bred in aquaculture). Several other fish found in the northern half of Australia are kept in captivity (e.g. archerfish, Hypseleotris gudgeons), but they're generally species also found in other countries.
Several of the Australian mudskippers are among the most widespread species and also found in large parts of south-east Asia; sometimes even throughout the Indo-Pacific region. Some of these are in captivity. However, I guess neither of the Australian endemic mudskippers (Boleophthalmus birdsongi and Periophthalmus takita) are in captivity outside the country and I'm not even sure if they're in captivity within the country.