Austin, 37, collapsed Monday afternoon in the city's Copeland Street fire house, and was pronounced dead at Quincy Medical Center, The Boston Globe reported Tuesday.
The state medical examiner's office will conduct an autopsy to determine the cause of death, officials said.
"Our entire community is deeply saddened by the loss of Firefighter Austin. It is a sad day for the city of Quincy," Koch said in a statement, adding an expression of condolence to Austin's parents.
Austin's death will be treated as a line-of-duty death, and flags in the city have been lowered to half-staff in his honor, officials said.
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