Dr Fox is due to meet Australian prime minister Malcolm Turnbull and New Zealand deputy prime minister Winston Peters to discuss current and future trade links after Britain breaks with the bloc.
However, his week-long trip threatens to be overshadowed by a dispute over the quota-splitting plan, which would see the UK and Brussels divide up the numbers of goods that can be brought in on low or zero tariffs based roughly on current rates.
Australia and New Zealand are among the countries to raise concerns over the move, arguing it would see a decline in the quota available and so harm exports.
Australian trade minister Steven Ciobo said it would impose unacceptable restrictions.
He told BBC Radio 4’s Today programme: “The point is that you have a choice about where you place your quota at the moment.
“Therefore, given that you could put it in the UK or you could put it into continental Europe, why would we accept a proposition that would see a decline in the quota available because of the Brexit decision?”
A Department for International Trade spokesman said: “As we leave the EU, we will need to update the terms of our World Trade Organisation membership to reflect an independent UK trade policy.
“We want to ensure a smooth transition which minimises the disruption to our trading relationships with other WTO members and tariff rate quotas are one of the issues that we are discussing with them.
“This is largely a technical process and we will continue to engage WTO members including Australia in an open, inclusive and transparent way.”
Against a backdrop of rising British exports to the Antipodes, latest figures show trade between the UK and Australia totalled more than £13billion last year, while with New Zealand it stood at £2.5billion.
The Department for International Trade’s analysis forecasts that 90 per cent of world growth in the coming years will come from outside the EU and the Government says it is seeking to position the UK to take advantage of this trend.
Dr Fox said: “The UK, New Zealand and Australia already have a strong economic and cultural relationship – based on our shared histories, language and open liberal economies. This visit highlights the commitment on all sides for this friendship to continue.
“Together with my colleagues from Australia and New Zealand I look forward to laying the groundwork for potential free trade deals, and looking at practical steps we can take now to boost trade and investment ties, while making it easier for our businesses to work with their partners on the other side of the world.”
As part of the week-long visit, Dr Fox will also meet trade organisations, business groups and UK firms.
He will also lend his support to Britain’s BAE Systems, which is bidding to secure a multibillion pound contract to build Australia’s next generation of frigates.
Dr Fox will be accompanied by the Government’s chief trade negotiations adviser, Crawford Falconer.