It refers to several things. First, it refers to the passion felt by Grace Ansley and Delphin Slade. Delphin and Alida were dating, but he has a bit of a crush on Grace. Alida senses this, and one night, when they are all in Rome (their families are well-to-do and travel in packs, it seems), Alida sends Grace a note signed by Delphin arranging a meeting. Unknown to Alida, Grace answers the note, and the two really meet. That night, Grace's daughter, Barbara is conceived.
Second, it refers to the actual fever contracted in Rome. This is the reason Alida sends Grace the note. She is secretly hoping that Grace will go to meet Delphin and contract the fever which caused many to die. Grace did get sick, but she also got pregnant. She recovered from the fever, and married her husband who reared the child as if she were his own.
Third, it refers to the fact that Alida is jealous of Barbara. She isn't sure why, but she wishes her own daughter, Jenny were more like Barbara. Jenny does not match up to Barbara's beauty, charm, and lively personality. When the two women are in Rome once again, talking on a veranda and looking over the Roman ruins, Grace smacks Alida with a Roman fever of her own--Barbara is Delphin's child. The secret is finally out after twenty plus years and the deaths of both their husbands. The joke is on Alida Slade.