The Refugee Action Collective said the 22-year-old woman known as Naima and her baby boy were flown in separate planes.
Ian Rintoul from the advocacy group said Naima underwent a caesarean operation on Nauru on Thursday and the baby was born one month premature.
He said an air ambulance transported Naima to Brisbane and a plane carrying the baby left Nauru three hours after his mother.
Dr Barri Phatarfod from Doctors for Refugees said the mother was on life support.
