In June there will be several festivals and fairs around Tuscany. Not wine oriented, because wine festivals take place generally after the harvest (September) when the novello wine is presented unless there are special events tied to wine at other periods of the year.
There will certainly be local food festivals (sagra) in the first two weeks of June. The only problem is that these kind of festivals aren't often promoted much ahead of time or advertised on main event networks.
You generally find billboards or posters a few days before the event. There isn't an official calendar with these festivals even if they are recurring annual events, dates are decided upon every year.
I strongly suggest you go to local tourism information offices once you are in Tuscany and ask them to provide information on local events. For example, if you intend traveling to Chianti you can stop in Greve in Chianti and ask there.
I also recommend you ask at your hotel. Locals know what is going on and generally can offer tips and suggestions.
Please don't hesitate to come back with any other question
Since you were looking for wine related events, take a look at http://www.classico-e.it/ - wine events hosted by the Chianti Classico consortium from May 29 to June 5 - I think it sounds like what you were looking for!!
I had forgotten about that, the Mostra del Chianti, also centered around the Chianti coming from around the Montespertoli area. The dates this year are from May 29 to June 6, almost same as other event. More info here: http://www.mostradelchianti.it/
And before I forget again, May 30 is Cantine Aperte, a sort of open house for wineries that participate in the event. The easiest thing to know which ones will be open is to ask at the tourist info point where ever you are, they'll likely have by then a handout with the ones open in the area.