In short - yes you can. A trader should trademark a business
name because it provides the right to exclusive use, added
protection from competitors, increased value of brand and goodwill
as well as adding value to the business itself.
It is a common misconception that registering a business name is
a good way to prevent others from using that name. Accordingly,
registering your business name with ASIC is not enough to protect
your rights to a name.
Trademark registration is a relatively inexpensive and effective
way to achieve protection of your business name. Upon registration,
trademark owners are afforded a bundle of rights relating to ways
in which they can use the mark to the exclusion of others. This
includes the right to exclusively use the mark in Australia to
brand your goods or services, licence or assign it as well as the
right to take legal action against other traders whom make
unauthorised use of your mark.