No, keeping a quokka as a pet is not possible, nor a good idea. These animals have razor-sharp claws that will cut you. They are also nocturnal and have a habit of raiding anything they can in the middle of the night for food. Quokkas also cannot be housebroken and require a ton of space to be happy.
Aside from all of this, the government will not allow you to have one as a pet. Simply touching a Quokka can easily result in a $300 fine. Take pictures with the cute animal in the frame, make your friends jealous, but don’t take one home!
Unfortunately, it is illegal to have a pet Quokka because it is a wild animal. They are ridiculously cute and need an ample amount of space in which to live. They cannot be housebroken, and they are nocturnal. They also tend to bite, and they have very sharp claws.
They are a vulnerable species. You could be fined for touching or petting a Quokka. Australia has stringent rules on the export of wild animals. It is even less likely to own a Quokka as a pet if you are not in Australia.