Do I need a referral to see a physio?
Short answer: no. Longer answer: a referral changes what it costs, not whether you can come.

You can book directly
In Australia, physiotherapy is a primary contact profession. That means you can make an appointment yourself, without seeing a doctor first, and without anyone's permission.
Most people who book privately do exactly this. You ring the clinic, you come in, you get assessed.
When a referral does matter
A referral matters when someone else is paying part of the bill.
If you have a long-term condition, your GP can write a GP Chronic Condition Management Plan. Many people still call this a care plan, a CDM or an EPC referral. It refers you to allied health, including physiotherapy, and Medicare then pays a rebate towards a set number of visits each calendar year. You still pay the difference unless the clinic bulk bills.
WorkCover and TAC claims need their own paperwork and an accepted claim number. NDIS participants who are self-managed or plan-managed can book without a referral, though your plan manager will want an invoice with the right support item on it.
Some private health funds also ask for a referral before they will pay an extras rebate. Most do not. Check with your fund.
What to do if you are unsure
If you already have a GP appointment coming up, ask them whether a care plan applies to you. It costs you nothing to ask, and if it applies it will save you money over a year.
If you are in pain now and do not want to wait, book in as a private patient and sort the paperwork afterwards. A care plan can be set up later.
General information only, and not a substitute for individual advice. If something is worrying you, see your GP or book an assessment.