No — suicide is not illegal in Australia.
Mondaq
+4
Criminal Defence Lawyers Australia
+4
Law Handbook
+4
Here's a breakdown of the relevant parts of the law:

What you can't be prosecuted for
You cannot be criminally charged for attempting suicide or for surviving one. It's not an offence in any state or territory.
Sydney Criminal Lawyers
+3
Criminal Defence Lawyers Australia
+3
Law Handbook
+3
You also can't be found guilty of murder or manslaughter for having attempted suicide or being part of a suicide pact (if you survive).
This is for Australia where I'm from, I'm not sure what the laws are elsewhere.