"And from then on, we've built an incredibly successful society, best multicultural society in the world."
Actually, Australia Day marks the arrival of the First Fleet in January 1788, which was commanded by Captain Arthur Phillip. Captain Cook had been dead for nine years at that point.
The English explorer and navigator first set foot in Botany Bay on April 29, 1770, about 18 years before the First Fleet landed at Sydney Cove.
But Senator McKenzie isn't alone in the misconception.
Greens senator Sarah Hanson-Young earlier this year sent out a statement attacking the government for funding a Captain Cook monument.
"Despite an important national debate about changing the date of Australia Day away from Captain Cook's landing at Botany Bay, the government has decided to spend taxpayer money it is stripping from the ABC on yet another monument to Captain Cook on the land of the Dharawal people," her statement read.
A spokeswoman later said the senator had not read the statement before it was sent out.
Australia Day has been held on various dates throughout the 20th century, but in 1994 it was celebrated as a national holiday for the first time.
Prime Minister Scott Morrison is calling for another day to recognise indigenous Australians, many of whom recognise January 26 as Invasion Day.