The short answer is no.

However, if you are the trademark owner there may be action you can take to acquire the associated domain name if owned by someone else in very specific circumstances. And, such action will depend on the type of domain in question, for example a .com or .com.au

Firstly - trademark registration is a country by country right - so if the domain is owned/used by someone in a different country to where the trademark is registered then you can't simply claim ownership to the domain because you own the trademark 'somewhere in the world'

Secondly, if the person is in the same country but uses/promotes the domain name for an unrelated business type to your own, and to the goods/services subject of the trademark registration then they're not necessarily do anything that breaches your trademark rights. 

Thirdly, if the person is in the same country and using the domain for exactly the same type of product/service to what your trademark registration covers but has used it for longer than you, then they may not be doing anything that breaches your trademark rights.

My tip on these sorts of things - or, anything 'infringement of trademark' related -  always do your homework and be certain of your rights (even if this means seeking professional assistance) before making any contact or threat against someone using a similar name/trademark. You can in fact get in trouble and be sued if you make a threat of legal action that's "groundless" so always careful.

To add to Jacqui's great answer, a trademark does not grant you eligibilty to a .com.au domain name just because you have that phrase or word trademarked.

There is no "heirarchy of rights" in .com.au and .net.au. In other words, just because you have a trademark, a business name, was around first, etc. etc. is irrelevant. What matters is:

- You registered the domain name before anyone else
- You have a valid ABN
- The domain name is closely and substantially connected to the website (i.e. www.fish.com.au actually sells fish, is a business called Fish, or perhaps is a family name, e.g. Fish Plumbing.)

If you let the domain name lapse for whatever reason, you can not simply "get it back" because you have a trademark using that domain name. You can complain to auDA, but unless the registrant breaches the domain name eligibility and allocation rules, they likely won't delete it.