A tympanic tickle was likely a type of sexual act Ferengi could perform on one another. When performed incorrectly, it could result in an infection in one's lobes. As the techniques involved were unique to Ferengi anatomy, most non-Ferengi were unaware of the tickle's existence; however, Federation doctors could cure the infection with antibiotics.
Quark once received a tympanic tickle from someone (likely a dabo girl) who lacked the necessary experience, and Dr. Bashir had to cure the resultant infection. When the doctor, who was unfamiliar with this term, asked what it was, he quickly answered his own question: "Never mind. I don't want to know." (DS9: "Afterimage")