Yes, you can sell an unregistered car in Queensland; it’s not illegal, just admin-flavoured.
You don’t need a Safety Certificate to sell it unregistered, but the buyer will need one (and CTP) to register it in their name.
Keep it off public roads unless it’s on a trailer or covered by an Unregistered Vehicle Permit; no permit, no road test, no “just around the block”.
Do a proper receipt with VIN, odometer, price, date, and both parties’ details, and hand over any keys, books and spare parts like a civilised human.
Plates shouldn’t be used on the road if the rego’s lapsed; the buyer handles registration with TMR using your receipt and their ID.
If in doubt, ring TMR and sleep soundly—paperwork beats penalties every day of the week.