While Q plates for cars ceased to be issued in 1955, existing ones at the time could still be used til the car died. Of course some still stayed alive. In modern times it became possible to transfer to another car by paying a fee to MRD, and from that time a pair of new Q plates had to be used ( in QLD back in the day only one plate was required and it went on the rear).
But for motorcycles it was different. The Q motorcycle plates not only ceased to be issued, they were recalled, and the new stamped steel plates were issued. That is why none exist, only in collections and museums.