Well, there is something wrong with it. It's not a Cadillac, it's a 1905 Locomobile Model C Roadster. (But, okay. "Geronimo's Cadillac" has a better ring to it than "Geronimo's Locomobile." I get that.)
Michael Martin Murphy himself has said that the inspiration for the song was the above photo showing Geronimo wearing a top hat and sitting at the wheel of a car. It was taken on June 11, 1905 at an exposition of some kind in Ponca City, Oklahoma. Don't know if the old chief actually drove the thing around some, or just posed behind the wheel, but it's a pretty striking visual.
Some Locomobile trivia: WWI General John J. Pershing owned one... Clive Cussler wrote the 1905 model into his 2010 novel The Spy... Rudyard Kipling hated his early steam powered Loco; called it "a nickel-plated fraud"... and the Locomobile was the first automobile to be used in warfare. (The Boer War in 1901. The Brits used it as a searchlight platform or something.)
Oh... The Indian in full headdress to Geronimo's left in the car photo is wearing a handsome beaded vest. Geronimo so liked it that the guy gave it to him, and when Geronimo died in 1909, he was buried in that vest.