Wooden items are allowed to enter Australia if they are free of bark, insects and signs of insect damage (such as piercing holes) or any other contamination. To check for insect damage, look closely at wooden items for holes and sawdust. Wooden items must be declared and inspected upon arrival. The short answer is yes, however, they must meet strict import conditions related to biosafety.
I have brought the prices of wood and fabrics without problems. They will check it carefully and return it to you if there are no problems. If you want to bring or ship a wooden product to New Zealand, you must declare it and possibly it will be inspected according to the nature of the item. All wooden items must be declared on your customs forms, whether you bring them with your personal baggage during the flight or on a cargo ship in a container.
There is no problem, or certification required, to ship or take wood products abroad, with the exception of New Zealand, whose Customs can inspect unfinished wood products if they deem it necessary. Some treatments of wood items can only be performed by an approved treatment provider; be sure to check what type of treatment your item needs and what acceptable treatment would be to ensure that it is accepted upon arrival in Australia. Think of all those happy tourists who leave the country every day clinging to their didjeridoos and wooden boomerangs, all confident to take them home to be given a place of honor among travel memories, without any problem. Wood with attached bark or insect holes or in an unfinished raw state will attract attention, while finished wood products will generally be allowed to pass.
Most people who bring wooden items or furniture to Australia will have treated items that do not pose a risk. If the wooden object you want to carry has been (or is) infested by some kind of insect, you have two options. It's a question often asked on emigration forums and Facebook groups, so here's what you need to know about shipping wooden items to Australia. For a complete and up-to-date list of wood items you can ship to Australia, see the Agriculture website here.
Regardless of how items enter the country, whether through a cargo ship, personal baggage or by mail, all wooden items must meet the same biosafety requirements. Customs reserves the right to open and inspect any or all of your belongings, not just your wooden or high-risk items.