A person from Melbourne is a Melburnian. The variant spelling 'Melbournian' is considered grammatically incorrect as well as making us mel-BORN-ian whereas we are mel-BURN-ian. Sometimes you'll see Melburnite (but they have no idea).
Local knicknames for people from Melbourne are Mexicans, W.A. and Perth are Sandgropers, Adelaide (South Australia) are Crow Eaters, Sydney (NSW) are Cockroaches, Brisbane (QLD) are Cane Toads, Northern Territory are Territorians, Darwinian from Darwin and people from Tasmania are called Tasmanians.