I'll be travelling to Australia soon and I plan to do some spearfishing while there. As a non-Australian, do I need to obtain a permit to spearfish in Australia?
It depends on the region and circumstances. For example, for NSW you do require a 'recreational fishing license', which is $7 for 3 days and can be bought online, or through various outlets including Kmart, but if you are fishing on a charter boat / hire boat or with a guide, they may already have a license which will cover you. (Reference)
Obviously you should also check what the local regulations in terms of allowable equipment, permitted areas, etc, as these might also vary. You should be able to find specific regulations for the areas you're planning to visit without too much trouble.