No. Registering a business/company name entitles you to trade under that name but does not give you a legally enforceable right to exclusive use and control, such that a trademark would confer. It is important to check that your proposed business name is not subject to a trademark or patent prior to registration by visiting IP Australia.
Should you choose to trademark your name, your intellectual property right to it will be protected from unauthorised use within Australia.
How do I register a trademark?
You can register your trademark by visiting the IP Australia website and completing an online application. You must meet eligibility requirements for your trademark to be considered. Aside from your status as a legal entity, i.e. individual person, company etc. these include significant difference from existing trademarks.
It can take 3-4 months for your application to be reviewed and there are costs involved.