Google Sydney's internships are indeed primarily intended at students in Australian universities. However, if you apply for a full-time job that's specifically listed as open in Sydney, and you get it, Google will pay relocation costs:

https://www.google.com/about/careers/search?jl=Sydney,Austra...

Disclaimer: I applied to work at Google Sydney, got the job, and was relocated. (From Melbourne, mind you, but there are people here from all over the globe.)