Generally, no. Hanging stockings is not a common custom among typical Australians. The closest most people get to a Christmas stocking is receiving one filled with chocolates on Christmas Day (or Christmas Eve, for some people).
Some Australians do put them up, but they are really not featured as much as they are in the US.
You hang Christmas stockings, with care.
People hang stockings so they can be filled with small treats on Christmas eve.
You hang stockings, with care.
it's a tradition
Stockings are hung on the hearth at Christmas.
to get more gift
it is a tradition
This is an individual family thing. Some children in Australia do hang stockings for Christmas; many do not.
Socks
Children typically hang stockings over the fireplace before Christmas night in the hopes that Santa Claus will fill them with gifts and treats.
If it is Christmas you hang stockings, decorations such as bells or ornaments. Other than Christmas you can hang pictures I suppose.
They hung stockings by the fire on Christmas Eve so that Santa could leave gifts.