It would be more surprising to me if there was a rich, noble Italian family in the 1200s-1500s that didn't have falcons or participate in falconry, it was so widespread and insanely popular...but in the interest of searching out some actual support for this, I've got a couple of online sources I found.

Here's a singular reference on page 438 of Early Modern Zoology: The Construction of Animals in Science, Literature, and the Visual Arts

Other European rulers regularly sent falcons to Spain as diplomatic gifts, as did the Duke of Brandenburg annually, and Catherine de Medici in 1560

If you look at the fulltext version of Volume I of Lettres de Catherine de Medici and search for the word "faucon" you can find two instances when she talks about falcons or falconry, but I can't make much of them as I don't speak French...

the first reference seems to be talking about Catherine trying to reduce a royal debt by sorting out finances in the King's house, including either reducing or axing the falconry group ("une partie de la fauconnerie") but again I am probably super wrong:

C'était là la grande difliculté. Le déficit que le précédent règne avait légué se montait à quarante-trois millions et demi. Pour rendre les Etats plus dociles, c'est en vain que Catherine avait tout à la fois sup- primé la vénerie, une partie de la fauconnerie, réduit la musique de la cha- pelle, et réformé presque en totalité la maison du roi, les députés reculèrent devant un si gros chiffre...

The second reference is either thanking the Duke of Ferrare or the Duke of Savoie for sending her some falcons with an envoy...I think.

Remercie le duc de Ferrare de ses bonnes démons- trations, i.3'i. — Le remercie de faucons qu'il a envoyés, i35. — Complimente affectueusement le duc de Savoie, i35.

However, solid evidence that at least one member of the Medici family had falcons and actively hawked can be found in Lorenzo De'Medici, Selected Poems and Prose by Lorenzo de'Medici, Jon Thiem

The first poem, The Partridge Hunt, actually describes a fictionalized story of a day of falconry with four falconers, of four different personalities, with four different hawks and their hunting styles. The description after the poem offers:

Hawking was Lorenzo's favorite sport, and the work remains an important documentary source for our understanding of this patrician pastime...

if you do a manual search for the word "falcon", you'll find a tiny excerpt from page 12, missing from the preview, reading:

Almost everywhere we encounter Lorenzo's love of the outdoors, his delight in the beauty of the Tuscan countryside and in the many pleasures it offered, from falcon hunting and swimming to observing the habits of shepherds, foxes, and owls...

There's also the fact that one of the emblems of the Medici family used to be a falcon holding a diamond ring, but that's not really evidence that they necessarily had falcons, since it seems to be more about the pun.

We need look no further than the popular Medici "impressa" of the falcon holding a diamond ring in one of its talons as an example of how they might be put together just on the phonetic similarity of the words chosen . The motto for this particular emblem of a Falcon e diamante was Fa con Dio Amante which, translated from the Italian reads, Do all for the love of God.

As for the girl who supposedly ate her favorite falcon, I haven't a clue and have absolutely no evidence, but for some reason that must be a theme because I swear I've heard reference to it before. It might be a semi-popular moral of the story thing or a way to comment on someone's character, or it might be an actual instance.

Anyway this is mostly born out of my brain going "hey you've read about the Medici family's falcons before" and then bumbling around trying to find out why I'd think that. I hope any of this is of any kind of use to you, but personally I think you should just find as many paintings of the family as you can and I'll bet you'll find some rich Medici's portrait of themselves holding their favorite falcon.