However, a tsunami warning has been posted for parts of Australia's east coast and Lord Howe, Norfolk and Macquarie islands.
Injuries on Tonga are currently unknown with disruptions to communications.
Mr Hunt said Foreign Minister Marise Payne and the Department of Foreign Affairs and Trade are working with Tongan authorities.
"DFAT will continue to engage with the Tongan Government and has offered through the foreign minister all possible support that may be required," Mr Hunt told reporters in Canberra during his latest COVID-19 update on Sunday.
The Bureau of Meteorology said the Hunga Tonga-Hunga Ha'apai volcano erupted at 3.10pm AEDT on Saturday, and tsunami waves have been observed as a result of the eruption.
The bureau said a tsunami wave height of 1.27 metres was observed on Norfolk Island at 9pm AEDT and an 82cm wave was registered on the Gold Coast at 10.54pm AEDT on Saturday.
Waves up to 1.10m-high were being recorded at Ned's Beach on Lord Howe Island about 11pm AEDT and a 50cm surge was observed at Hobart's Derwent Park about 11.44pm AEDT.
Port Kembla in NSW's Wollongong registered a 65cm wave at 2.50am AEDT on Sunday.
Countries around the Pacific were also on alert, with the southern Japanese island of Chichijima recording a 75cm wave at 1am AEDT.
The bureau earlier detected a 1.19m wave in Nuku'alofa, Tonga's capital.
Land warnings were issued earlier for Norfolk Island and Lord Howe Island, however by 10:30am on Sunday these were downgraded and replaced with marine warnings.
"In areas with a threat to the marine environment only, emergency authorities advise people to get out of the water and move away from the immediate water's edge of harbours, coastal estuaries, rock platforms and beaches," the bureau said in a statement.
BOM meteorologist Sarah Scully said these warnings would remain in place until there were "no significant observations for six hours".
"We're waiting to be really sure that the impacts and the energy from the volcanic eruption has dispersed and it's no longer a danger to any people out of the water," she told ABC News.
Marine warnings are current for all coastal areas of NSW and large parts of the Queensland, Victorian and Tasmanian coasts.
In Queensland the marine warning is in place for Sandy Cape to Point Danger including the Fraser Island coast, Sunshine Coast waters, Moreton Bay and Gold Coast waters.
In Victoria a marine warning is in place from Lakes Entrance to 60 nautical miles east of Gabo Island including the East Gippsland coast.
The marine warning also covers Macquarie Island and parts of Tasmania including the northern tip of Flinders Island to South East Cape, east of Flinders Island, Banks Strait and Franklin Sound, the upper and lower east coast, the southeast coast, D'Entrecasteaux Channel, Derwent Estuary, Frederick Henry Bay, Norfolk Bay and Storm Bay.
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