The character you are referring to is Harry Powell, played by Robert Mitchum, who is not actually French, but presents himself as one. He travels to West Virginia, not India, searching for a couple's hidden loot that they stashed away before their execution. Harry believes the money is hidden in a place called "the place where the birds sing", hence his obsession with the name "Nightingale".
While the movie is not set in India, it's understandable why you might associate it with a Frenchman searching for "Mr Nightingale" due to the specific details of the character and his motivations.