The incident took place on Canal St between Bourbon and Chartres streets at about 03:20 local time (09:20 GMT).
Police said on their Twitter feed that “one suspect had been apprehended near the scene”.
They later said the person’s possible involvement was still under investigation and that no arrests had yet been made. No other details have been given.
Video footage from the scene showed numerous police vehicles cordoning off the area as forensic teams made checks.
He said: “Unfortunately, there were so many people out here we were unable to determine who was actually firing shots at the time. We do not know how it started.”
The French Quarter has been hosting holidaymakers marking the weekend after Thanksgiving.
Thousands of fans and alumni have also been drawn to the city for the Bayou Classic football game traditionally played on Thanksgiving weekend between Southern University and Grambling State University.
In June 2014, another shooting incident on Bourbon St left one person dead and nine injured.