Authorities said the man landed at Melbourne Airport between 6.20am and 9.50am before taking Qantas flight QF610 to Brisbane.
People who were at Melbourne Airport during this time have been urged to monitor for symptoms until July 21.
Queensland Health said the person also visited several locations in Brisbane while infectious.
Metro North Public Health physician Dr Megan Young said anyone at these locations during these times should monitor themselves for symptoms for 18 days from when they may have had contact.
“If they do develop symptoms, they should seek medical advice,” Young said.
“Anyone seeking treatment for an illness they think could be measles should call ahead before visiting their GP or medical practice to allow precautions to be taken that can prevent the illness spreading to others.”
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The illness usually begins with cold-like symptoms such as a runny nose, red eyes and a cough, followed by fever and a rash.
The characteristic rash usually begins three to four days after the first symptoms, generally starting on the face and then spreading to the rest of the body.
Measles spreads quickly with close contact, especially in those who are not fully vaccinated.
Young children and adults with weakened immune systems are the most at risk of serious illness.