I don't think so in Godfather I and I am not sure he is later mentioned in any film. If it is made clear one way or the other in the book or subsequent books, that would be interesting.
There is no information on Fontane discovering precisely what the Don did to bring Woltz around.
Indeed, Woltz went to great lengths to make sure that no-one would find out.
Woltz gave the necessary orders. His personal confidential staff swung into action. The servants and the doctor were sworn to secrecy on pain of incurring the studio’s and Woltz’s undying enmity. Word was given to the press that the racehorse Khartoum had died of an illness contracted during his shipment from England. Orders were given to bury the remains in a secret place on the estate.
The Godfather - Mario Puzo.
In answer to your comment, in the novel Woltz does try to mess with Fontane's career after the movie is completed but I do not believe that is covered in the movie as Fontane has a much reduced role compared to the novel.
Hagen lit a thin cigar. “We got the word that Jack Woltz won’t spend studio money to support your candidacy [for an Oscar]. In fact he’s sent the word out to everybody who votes that he does not want you to win. But holding back the money for ads and all that may do it. He’s also arranging to have one other guy get as much of the opposition votes as he can swing. He’s using all sorts of bribes — jobs, money, broads, everything. And he’s trying to do it without hurting the picture or hurting it as little as possible.”