No. Such constructions were the norm in Old English and there's no reason to assume French influence was necessary for them to be available to Chaucer's Middle English versification.
Middle English, the flavor of English used by Chaucer, has deep continuities with Old English, the version of the language that Beowulf is written in. Old English is a highly inflected language, where number, person, gender, case, etc. all affect the morphology of nouns. In highly inflected languages, since a word's form does reveal its grammatical function, word order itself does not matter very much. Consequently, Old English poetry has plenty of examples of word order we would find unusual in Modern English. E.g., here are some lines from the account in Beowulf of the ship-burial of the king Scyld Scēfing:
... him on bearme læg
mādma mænigo, þā him mid scoldon
on flōdes ǣht feor gewītan. (ll. 40–42)
Slade, Benjamin. Beowulf: diacritically-marked text and facing translation. heorot.dk. Accessed 23 Feb 2021.
Word-for-word translation (mine):
... him on breast lay
treasures many, that him with should
on flood's domain far go.
Idiomatic translation (mine):
On his breast lay many treasures that would travel far out on the sea along with him.
The quotation has many examples of word order that are unusual in Modern English: him mid / him with; feor gewītan / far go; and the specific example you're asking about, with adjective following the noun it modifies, mādma mænigo / treasures many.
As David Crystal notes, the transition from Old English to Middle English did involve a loss of inflection, and consequently, a valorizing of word order rather than morphology to determine semantics. And it is true that Middle English vocabulary and syntax was heavily influenced by Norman French. That said, Middle English still is English; its progenitor is Old English, not French. To Chaucer, the word order shoures soote would be available because of the enduring influence of Old English and not solely dependent on the more immediate presence of Norman French.
Finally, word order in poetry is a matter of craft as much as of syntax. Even if Old English had not had flexible word order, it's not unreasonable to attribute Chaucer's anastrophe to his desire for a rhyme with roote in the next line rather than to the influence of French.