What was the profession of 1st Greeks who arrived in Australia and became famous for that?

You’ve read something somewhere, OP, I can tell, but I’m at a loss about where. The answer, pace Romain Bouchard, is not in Wikipedia, but I don’t remember it.

Let me try and reconstruct it.

The big Greek migration wave into Australia was in the 1950s–70s. The stereotype was milk bar owner (= grocery story) or fish and chip owner, because that was highly visible (my parents were among them). Less visible, but I suspect more common, was factory worker.

Milk bars are now just about extinct, and the fish and chip shops have bifurcated: some boutique nouveau fish and chip shops are Greek, but the common hole in the wall kind of places are now Chinese. As for factories, like much of the First World, we don’t have any any more. So the new wave of migrants from Greece (fleeing austerity and connecting with relatives in Australia) seem to have ended up here in Greektown Melbourne, mainly in the service industries.

This is not the time frame OP is asking about.

There was a Greek community worthy of the name since I guess the 1890s; the first Greek church in Melbourne was founded in 1900.

But I think the trade OP is alluding to is sponge divers. There were sponge divers in the north of the country, a trade practiced in Greece in the Dodecanese; that trade also accounts for the sizeable Greek community in Tarpon Springs, Florida. The pearling company Paspaley was set up in the 1920s by a Greek Paspalis family.

I do know that the pre-War Greek population of Australia are disproportionately from a few islands—Castellorizo (including the Paspaleys), Kalymnos, Ithaca in Melbourne. Castellorizo and Kalymnos are both sponge diver islands.