Rumors of incest between Lucrezia and her father and brother have dogged the Borgia family since Lucrezia's first marriage. Are they true? No one knows for sure, but probably not. You can get the whole story of how the rumors got started and why they aren't credible in A Borgia Daughter Dies, available at Amazon and Smashwords.
This is documented: the rumors were started by Lucrezia's soon-to-be-ex first husband, who was very angry with the Pope for accusing him of impotence in order to grant Lucrezia a divorce. (Non-consummation of marriage was one of few available grounds for divorce then.) Hubby #1 wrote a letter to a cousin stating that the Pope "wanted Lucrezia for himself"–a statement as ambiguous in Italian as it is in English– and the rumors went viral from there. Soon Lucrezia was rumored to be having sex with both her father and brother. Roman poets, the yellow journalists of the day, wrote doggerel about Borgia incest and sold it on the street, thanks to the newly-invented printing press. (Things like duels and libel laws weren't around yet.)
A mysterious baby fueled the rumors, too. Lucrezia hid herself for months in a convent while her first husband and father fought over the divorce, and likely had a baby there. The pope and Cesare successively declared themselves the father of a baby "by an unknown mother," which led their enemies to declare that Lucrezia was confused about who the father was. This is pure malice—the Borgias were way too smart for that. The pope and Cesare acknowledged fatherhood to give this child–-who obviously had a very important mother, whoever she was—-rights to financial support and inheritance. The pope was close to seventy, so it made more sense to have Cesare assume these obligations, which he did. Lucrezia herself cared for this "half-brother" from afar her whole life.
Why is Lucrezia the likely mother? First, she spent a lot of time hiding in a convent. Second, the bodies of her maidservant and a handsome young man who had visited while she was in the convent were found in the Tiber, tied up and stabbed. And before that, a prominent individual witnessed Cesare chasing this handsome young man down the hallways of the Vatican and stabbing him as he clung to the pope's ankles, begging for mercy.
But the young man was not a former groom named Paolo, the fictional "daddy" in the Showtime series. If you want to know who he really was, read A Borgia Daughter Dies.
Here are more reasons incest is improbable: Cesare and his father had plenty of sexual outlets and didn't live in the same place Lucrezia did. The pope was close to 70, fat, running the papacy hands-on, and absolutely besotted with "La Bella Giulia," his young mistress who was supposedly the most beautiful woman in Italy. How did he have time for incest, much less the energy?
As for Cesare, who had multiple mistresses and a number of acknowledged bastards: he had already contracted syphilis by the time this baby was born. But there is no evidence that Lucrezia ever had syphilis—she had miscarriages, but also a number of healthy children. She also lived much longer than Cesare did.
Incest? Very unlikely.