Yes they did, and for the same reasons.
Europeans lived in close proximity to livestock, with fairly poor hygiene and waste management, and their diet was fairly lacking in many important nutrients. These are issues with urbanisation, and thus Asia had the same problems too, and Eurasian trade spread these diseases far and wide across the continents. Pandemics have swept Eurasia for millenia. For European invaders, these issues were exacerbated by long sea voyages to Australia and near starvation in Sydney due to the alien environment and a lack of skilled farmers,
Indigenous Australians did not have these issues. They had no livestock, extremely healthy diets, almost no contact with urban civilisations and were semi-nomadic, meaning their waste was never going to make them sick.
Diseases like smallpox are believed to have been brought by Arthur Phillip's First Fleet to Sydney in 1788. Tribes had no immunity and no experience with pandemic diseases, so they feared it was evil magic and warred with their neighbours or fled, which spread it further inland. When European explorers crossed Sydney's Blue Mountains decades later, they found evidence of mass death in tribes that no European had ever had contact with.
Some argue that disease may have been brought by Indonesian Makassan fishermen, around the same time as Britain's invasion, but this is unlikely, as it would require seriously sick people to cross Australia's deserts. Others argue that diseases were released deliberately by persons unknown in the Sydney garrison, a biological weapon which has precedent in British experiences in North America, where some British soldiers in the First Fleet would have served.
Europeans pushed Indigenous Australians to adapt to European living, in an attempt at 'civilising' them, or pushed them to flee by warring or stealing their land. Both caused severe disruption to food gathering, adding to the effects of disease, killing skilled gatherers, elder teachers and the next generation of children. Their society disintegrated as their people died, their food and land stolen and their knowledge not passed on. People adapting to urban living didn't understand hygiene control, having never had to worry about it in the past, and made themselves sick.
Some Europeans took pity, and others bribed them away from settlement, by offering rations to the sick, encouraging more interactions with Europeans and sick tribesmen, spreading disease further. The foods Europeans offered, mostly flour, sugar and tea, were significantly less nutritious than native foods, but the desperate survivors had little choice but to rely on freely offered food, and poor nutrition made them more likely to fall ill.
The fact that Australia was settled slowly, in different locations isolated by extreme geography, meant that there were Indigenous nations being newly exposed to foreign disease well into the 20th century. In my area of Australia, the Noongar people of the south-west suffered repeated epidemics over one hundred years, from the 1820s to the 1920s - well after disease had decimated the eastern coast, where Europeans had landed in 1788. A century of being driven from towns, rape, massacres and exposure to disease meant that the survivors had developed immunity through European admixture or developed immunity. Some desert tribes were not discovered until the 60s and 70s - these were lucky enough to escape epidemic thanks to modern medicine and hygiene, and severe isolation.
That being said, Indigenous Australians, whether they are urban or regional, still suffer from disease at far higher rates than non-Indigenous Australians, mostly due to the effects of poverty and lack of services in their area.
Some good sources:
The Rise and Fall of Ancient Australia, by Geoffrey Blainey. (A long, dense read)
A History of Australia, by Peel and Twomey. (A quick chapter on it)