Doesn't seem like it. The FitzGeralds' originator was a Norman from Wales, Gerald FitzWalter, who as the name implies was the son of a man named Walter, Walter FitzOther, who is in the Domeday Book. His father would've been named Othere, which is an old Anlgo-Saxon version of the Norse name Óttarr. Anyway; so FitzGerald is a fossilized Norman patronymic which by all appearances became a family name when they arrived in Ireland in the 12th century. This doesn't make for any obvious link to Italy, where the Gherardinis on the other hand had been established for some time by then.

This story (Italian) claims that unspecified Italian chronicles made the connection. I can't verify that, but I wouldn't doubt they do say as much. However, for the same reason the reliability would have to be considered pretty low. The reason being that medieval chronicles tend to be full of exactly those of folk-etymological story-telling to explain links between people with similar names. (the Gesta Hungarorum of Simon Kézai connected the Hungarian word concluded they'd once conquered Spain on the basis of the resemblance of 'hispania' to the Hungarian word íspan. The Getica by Jordanes equated the Goths with the Getae of Thrace with the Geats of Sweden. The Prose Edda from Iceland equated Thor with Hector and Loki with Ulysses on the flimsiest of name similarities; I could go on and on..)

So the family name is absent and it's odd to say the least that it would be abandoned, turned into a first name, then patronymic to be re-adopted as a family name. Yet another problem is that these are not actually the same name. There's an 'l' differing.

Gherard (and thus Gherardini) is from the Germanic name Gerhard (and variants) which is from ger meaning 'spear' (remains a element in words like 'garlic' or 'auger') and hard meaning hard/strong. The equally-Germanic name Gerald, on the other hand, shares the same first element but the second part is wald (as on 'Oswald' or even 'Arnold') meaning power or strength (even in modern English, although it's dialectal)

One might reason (as that article did) that the Anglo-Normans simply corrupted the name as they couldn't pronounce it. It's not very obvious why that would be the case; not only does the name Gerrard exist as a native variant of Gerhard in the British Isles ; it was introduced there by the Normans themselves, like many other names of Germanic origin (e.g. William and Roger - there's that ger element again, BTW) It's more unexpected to find the name Gerhard/Gerard in Italy than in Normandy.

As for the alternate hypothesis, it's not hard to see how an Italian noticing the similarity between their own "Gherardini" (the descendants of Gherard) and the and the prominent "FitzGeralds" (son of Gerald) could get the idea that they were related. The very unusualness of Gerhard/Gherard in Italian contexts works in its favor here. So a later Italian source having a story to that effect when the contemporary Insular sources say something else doesn't really mean much. Add to that the fact that the names aren't actually the same and that the circumstances don't really make sense.

I doubt I'll shake any family traditions here but absent any real, documented, relation it seems more likely to be a bit of folk-etymology.