Is “ no news is good news” of Italian origin?

Someone has to venture an answer, even if it is sketchy. There is circumstantial evidence.

French?

First, the proverb also exists in French

Pas de nouvelles bonnes, bonnes nouvelles.

Very little is readily available on its origin, except a (non-scholarly) page that indicates, too, its Italian origin. So let's pursue that trail.

Italian: Proper Latin Form

Yes, the proverb is referred to in Italian but, interestingly, not in the form generally offered in English grammars:

Nulla nuova, buona nuova.

In Italian, it is used in its authentic Latin form.

Nulla nova, bona nova.

Indeed, the Latin proverb was sufficiently in common use in Rome, to have been carved on some monument (if someone cared to obtain the source indicated, we would know more about it).

Italian: Problem with the form quoted in English books

Problem: nulla nuova is not considered correct Italian, since the Latin meaning of "nullus" (no, as is no news) was not carried forward into modern Italian! The meaning of nullo, is rather as "void, invalid, null".

Questa decisione è nulla.

"This decision is void".

And that explains why a high-school grammar book of 1887, Regole ed esercizi di grammatica italiana per le scuole secondarie wanted to warn high-school pupils (p. 87) that nullo:

per gli antichi, valse nessuno : nullo male fece (nessun male fece); ma ora vive, in tal senso, solo nel proverbio: « nulla nova, bona nova »;

Translation:

meant, for the Ancients, none/no one: he did no [nullo] harm; but today it persists only in the proverb: "nulla nova, bona nova"

And if the proverb was mentioned in a high school textbook, it also suggests that:

a. this issue of "nullo" versus "nullus" was a common grammar issue in standard Italian [1887: a few decades after the unification of the country, when there was a strong drive to impose its use].

b. the Latin form was in wide use and was the proper one.

So why could English educated people refer to, in the seventeenth century:

nulla nuova, buona nuova

as allegedly "Italian"?

The best explanation seems phonology and more precisely vowel breaking. (Florentine) Italian tended to break a consonnant + 'o' like nova/bona into diphtongs: nuova, buona.

Hence, presumably, this odd Italian form of the proverb was a direct transposition ("Italianization") of the Latin form, in a process akin to folk etymology. How much this folk etymology owes to Italian speakers (i.e. this Italianized form was indeed common at some point, before being ruled out as "incorrect"), or how much it was inflated by the perception of educated foreigners, would be a subject of further study...