Maxwell is from New Zealand but has been living in Tanah Merah, a suburb of Logan City in Queensland, the Courier Mail reported.
A second man Armstrong Renata, also 21, has also been charged.
Maxwell's legal aid lawyer Tim Clements detailed the allegations against his client in court.
He said Maxwell was alleged to have approached Mr Miller and his friend and asked "if they wanted to fight".
"He is alleged to have thrown a punch at each of them", Mr Clements said.
The Courier Mail reported that police had not yet revealed who threw the fatal punch - Maxwell or Renata.
"It's alleged his co-accused has come from behind or the side and that is the punch which has led to (Mr Miller) falling to the ground," Mr Clements told the court.
Both men are due back in court on February 8.
The Courier Mail reported that Maxwell moved to Logan from Melbourne three months ago and is living with his cousin and uncle.
His parents live in Auckland.
Renata's lawyer Mitch Cunningham, said Mr Renata was "very upset".
Mr Miller's father Steven Miller told reporters his son suffered "massive brain trauma" and was surrounded by loved ones when life support was switched off.
"He struggled bravely in the intensive care unit... and our family and many close friends have been by his side since the incident," News.com.au reported.
"Cole was a beautiful, brave young man with his whole life yet to be lived. The whole Miller family would like to thank his school friends from Brisbane State High and the water polo community, and the many, many others who have sent messages of love and hope."
Mr Miller was a talented young athlete and had represented Queensland in under-20s water polo. He is the younger brother of Australian water polo player and Olympian Billy Miller.
"This is a random act of violence - there is no indication that the victim knew who the offenders were," he said but did not rule out that the two may have crossed paths in a nightclub earlier in the night," said Detective Acting Inspector Tom Armitt.
