The British two-time heavyweight champion, 36, is said to have sustained minor injuries as a passenger in the vehicle.
Two people were killed in the accident, which occurred along the Lagos-Ibadan expressway in south-west Nigeria.
Joshua was among four people travelling in a Lexus SUV when it reportedly collided with a stationary truck near a petrol station.
Footage on social media appeared to show Joshua inside the back of the damaged vehicle and wincing in pain.
He has since been taken to hospital for treatment, according to local state police.
Further video clips showed two bodies strewn across the road following the wreck.
An eyewitness at the scene told Punch newspaper: "It was a two-vehicle convoy: a Lexus SUV and a Pajero SUV. Joshua was seated behind the driver, with another person beside him.
"There was also a passenger sitting beside the driver, making four occupants in the Lexus that crashed. His security detail was in the vehicle behind them before the crash.
"Other eyewitnesses and I began the rescue and flagged down oncoming vehicles to assist. A few minutes after the crash, officials of the Federal Road Safety Corps arrived."
The incident comes days after Joshua knocked out YouTuber-turned-boxer Jake Paul in six rounds in Miami, Florida.
