Mr Moon and First Lady Kim Jung-sook arrived in Canberra on Sunday night ahead of a formal welcome and meeting with Prime Minister Scott Morrison on Monday.
Australia and South Korea are marking 60 years of diplomatic relations this year.
Mr Morrison said the relationship continued to grow, underpinned by trade, shared values, common regional strategic interests and a commitment to an open, inclusive and prosperous Indo-Pacific region.
"During President Moon's visit we will formally elevate the relationship to a comprehensive strategic partnership, reflecting the depth and breadth of our cooperation and people-to-people links, and common aspirations to work even more closely together in our region," Mr Morrison said.
Australia has comprehensive strategic partnerships, or their equivalents, already with ASEAN, China, Fiji, France, Germany, India, Indonesia, Japan, Malaysia, and Papua New Guinea.
South Korea is Australia's fourth-largest trading partner and fourth-largest export market.
Australia's major exports include iron ore, coal, natural gas and beef, and its major imports from South Korea include cars and refined petroleum.
A free trade agreement has been in force since December 2014.
The Korean government has plans for a "Korean New Deal" to aid the country's COVID-19 economic recovery, focusing on digital infrastructure to drive future economic development and green initiatives to reduce carbon emissions.
Mr Moon, who concludes his presidential term next year, has previously flagged greater cooperation with Australia on low-carbon technology and space development, as well as national infrastructure and defence.
In particular his country is pushing for a region-wide partnership to boost the production and distribution of hydrogen, which by 2050 is expected to account for 13 to 18 percent of world energy.
Korea is the first country to succeed with the mass production of hydrogen cars.
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