The comte de Vaudrec is mentioned roughly a dozen times in the novel.
He appears for the first time in Chapter 3 of Part I: George Duroy is making a visit to Mrs Madeleine Forestier when Vaudrec enters the room without having been announced. Mrs. Forestier seems to be embarrased "for a second", then introduces the men to each other. The second occurrence follows almost immediately after the first: Duroy thinks back to the elderly rich man without being able to put him out of his mind; he even imagines that the other man had been displeased to see him with Mrs. Forestier.
Vaudrec is mentioned again in Chapter 4, which describes how Duroy accomanies his colleague Saint-Potin to learn the basics of journalism. Saint-Potin claims that Mrs. Forestier is Vaudrec's mistress and that the count provided her dowry when she married.
Vaudrec is not mentioned again until the end of Part I (chapter 8): Duroy and Madeleine Forestier have held wake for Mr. Forestier, and Duroy is already weighing up his chances of marrying the newly widowed woman. At this point he thinks back to the rumour that Vaudrec had provided Mrs. Forestier's dowry (apparently without remembering he heard it from his colleague).
Vaudrec makes another appearance at the home of Madeleine, now Mrs. Duroy, in the second chapter of Part II. George Duroy has bought a bunch of roses on his way home to surprise his wife, only to find out that there is already a vase with roses on the chimney because Mrs. Duroy is expecting the comte de Vaudrec for dinner. George suppresses his suddden urge to throw away the roses he has bought. His wife wants the count to continue the weekly visits and hopes that the two men will get on good terms, which is also what happens. After Vaudrec has left, Mrs. Duroy describes him as a true and faithful friend.
Vaudrec is mentioned again in the same chapter when the narrator says that député Laroche-Mathieu comes to dinner every Tuesday, one day after Vaudrec.
Vaudrec is briefly mentioned in chapter 3 of Part II in a context that does not appear to have any relevance to Mrs. Duroy.
He is mentioned again twice in chapter 5 of Part II, in which we learn that he is dying. Madeleine is very upset by this news; she goes to his house, and she and a nephew of Vaudrec's stay with the count during his final hours.
The next day (chapter 4), Mr. and Mrs. Duroy learn that the comte de Vaudrec has left almost his entire fortune (worth 600.000 francs) to Mrs. Duroy. George wants to know whether his wife had been the count's mistress, which she denies. The count's decision to leave his fortune to her had already been documented in his original testament, which was written when she was still Mrs. Forestier.
George Duroy's suspicion reflects his own lifestyle: soon after their marriage, he had already asked his wife whether she had made her first husband a cuckold, but his wife denied it. Duroy himself cheats on his wife with two other women; in fact, there is even a scene in which one of his mistresses accuses him of cheating on her with another mistress. In the penultimate chapter, Duroy and the police catch his wife and Laroche-Mathieu (now a minister) in the act and the journalist uses this to demand a divorce. It never occurs to him that he punishes his wife for something that he takes for granted as a man; he never reflects critically about his own adultery. It is probably for this reason that an illicit relationship is the only reason he can see for Vaudrec's testamentary decision.
However, there is another possibility, which was already suggested by Saint-Potin's statement that Vaudrec had provided Madeleine's dowry. A dowry is usually provided by the bride's parents.
Chapter 1 of Part II mentions that Madeleine did not know her father:
She recalled her own mother, of whom she never spoke to anyone—a governess, brought up at Saint-Denis—seduced, and died from poverty and grief when she, Madeleine, was twelve years old. An unknown hand had had her brought up. Her father, no doubt. Who was he? She did not exactly know, although she had vague suspicions.
Most likely, Madeleine suspects that Vaudrec is her biological father. This would explain why the novel never has Duroy try to catch Madeleine and Vaudrec together in bed. On the other hand, the text never gives the reader certainty, which would have been easy enough by letting the count leave a letter (e.g. as part of his will, stating that the letter must only be opened by Madeleine) confirming his paternity. The novel does not even let Madeleine make her suspicion explicit to her husband, thereby leaving the relationship between her and Vaudrec as a puzzle for the reader to solve.