
You may not recognise this little building as a mosque, but it was actually built as a place of worship by Afghan camel train drivers brought to Australia to service Outback Australia. Its located in Bourke NSW.
The first Muslims of this type were brought to Australia in 1860.according to Hanifa Deen's historical account:
Quote: "On 9 June 1860, three ‘exotic-looking’ Afghan camel men landed at Port Melbourne
from Karachi with twenty or so animals to participate in the (ultimately tragic) Burke
and Wills expedition organised by the Royal Society of Victoria. The ill-fated journey,
however, proved the merit of camel over horse and bullock. Over the next ten years,
British entrepreneurs imported significant numbers of camels with their handlers, and
more animals were unloaded at Derby and Fremantle in Western Australia, Port Augusta
in South Australia and Port Melbourne."
( Reference: http://www.slv.vic.gov.au/sites/default/files/La-Trobe-Journal-89-Hanifa-Deen.pdf )
The first mosques were basic utilitarian buildings as clearly indicated by this next image:

The image shows a mosque at Maree (South Australia) in 1884. It could be the first mosque built in Australia, probably in the early 1860's.
These building (and the builders) were not the first Islamic contacts with Australia,
Australia's nearest neighbour, Indonesia, had gradually converted to Islam after Islam was brought into Indonesia by Sufi traders from Gujarat, India during the 13th century (CE) and by the 16th century, Islam had surpassed Hinduism and Buddhism as the dominant religion of the peoples in the Indonesian island chain.
By the early 18th century, Muslim Macassan traders from Indonesia were visiting the N.W. coast of Australia. Local indigenous art and language attests to their visits. If they built places of prayer or mosques, nothing has survived.
What I've quoted here comes from an academic study by Sam Bowker, a lecturer at Charles Sturt University.
The full essay was posted in 'The Conversation'
You can access it at: http://theconversation.com/friday-essay-the-australian-mosque-65101?utm_medium=email&utm_campaign=Latest%20from%20The%20Conversation%20for%20September%2023%202016%20-%205655&utm_content=Latest%20from%20The%20Conversation%20for%20September%2023%202016%20-%205655+CID_0fd9ae7608de675efc856f0fa3af9746&utm_source=campaign_monitor&utm_term=Friday%20essay%20the%20Australian%20Mosque