First step is to ring your local rego authority (RMS, VicRoads, whoever), and ask them if they have any record of it.
If it's been rego'd anywhere in the country in the last 10-15(?) years, it will be on the national database.
If that doesn't work, then ring the other state rego authorities and see if it is on their old databases.
If that doesn't work, then try Suzuki Australia. I've seen a couple of letters from Yamaha Australia saying 'this vehicle was sold with a compliance plate', which is enough for the rego people. I assume Suzuki will be similarly accommodating.