Sort of, but technically "no".
It seems that -i was sometimes used as a genitive singular ending for masculine names from Greek that end in -ēs in the nominative. But although many of these come from Greek first-declension nouns, they typically seem to be classified as third-declension rather than first-declension nouns in the context of Latin.
In fact, genitive singular forms in -i existed even for some masculine names ending in -ēs that weren't from Greek first-declension nouns, such as Diomedes (from Greek Διομήδης, Διομήδους) and Hercules (from Greek Ἡρακλῆς, Ἡρακλέους, but not directly: the direct source of Latin Hercules is supposed to have been an Etruscan form "Hercle"). Varro mentions the variability between -i and -is in the genitive singular of these two names in De Lingua Latina, Book 10, §49.
Even though this irregular use of Latin genitive -i looks suggestively similar to the Greek use of -ου as a masculine first-declension genitive ending, I can find no evidence that -i was ever viewed as a first-declension ending in Latin.
Grammars that mention genitive singular forms in -i for names in -es
Allen and Greenough §52a says that "Many names in -ēs belonging to the 3rd declension have also a genitive in -ī", giving the example "Thūcȳdidēs, Thūcȳdidī". This grammar classifies Thūcȳdidī as a second-declension form, and Thūcȳdidēs as a third-declension form.
Grammar of the Latin Language, by C.G. Zumpt (1836), says the following:
"the greater part of Greek nouns in ης, ου, if not patronymics, are declined after the third declension in Latin, as Alcibiades, Miltiades, Xerxes" (Sect. VII.3, pp. 11-12).
"Cicero very commonly, and other authors of the best age occasionally, form from Greek nouns proper in es a genitive in i, instead of is; e.g. gr. Achilli, Ulixi (see p. 15), Isocrati, Archimedi, especially from those in cles, Agathcli, Diocli, Neocli, Procli, Pericli, Themistocli. So in the barbarian names Mithridati, Xerxi, and others." (Sect XIII, Remark 1, p. 25-26).
In a comment, Draconis mentioned finding a single example of genitive singular Thūcȳdidī in Quintilian: "imitator Thucydidi et ut multo infirmior".
Side note: I have asked a separate question about the use of third-declension forms like
gen. sg. Thūcȳdidis for names like this: How are "Arsaces" and "Gotarzes" declined, and why?