A linguist may elaborate, but just to provide a brief answer - no, it appears to be simply a coincidence.

The English name "Austria" is in all likelihood a derivation of the German name for the country: Österreich (pronounced somewhat like "Austerraih"), meaning "eastern realm". There may also be an element of the Latin word "Marchia", meaning march/borderland to it, but the main origin is Germanic and refers to the east.

The name "Australia" meanwhile derives from the Latin word "Australis" meaning "southern". This derives from "Terra Australis" (South/Southern Land), the name for a landmass which from the late medieval period onwards was speculated to exist as a means to "balance" the northern Eurasian landmass.

Thus "Austria" is Germanic and refers to the east, "Australia" is Latin and refers to the south. Two different language families - two different cardinal directions.