Not any particular one that stands out in the area of Greve, but there are several events in the Florence surrounings.
In Florence, for example, on October 8, there will be a historical procession from Piazza della Signoria to Piazza Duomo carrying special offerings to the Cathedral in honor of Santa Reparata. A ceremony and mass will take place inside the cathedral starting at 6 p.m.
If you're interested in funghi (mushrooms), head toward Bagnolo in Grosseto on the Monte Amiata for a food festival based entirely on the fungo Amiatino on 10-11 October.
If you'd like to see an antiques fair, October 10-11, there will be fairs in both Pistoia and Siena.
Most local events are publicized locally so make sure to ask where you're staying at what is going on in the area, they'll be the best informed!