I cannot prove it with hard facts or numerical data, but I am Belgian, and quinze jours is the dominant phrase here as well for "two weeks", and not quatorze jours. If there was a regional difference in the past, it seems it has disappeared nowadays, and the usage in Belgium is the same as in France.

Some examples taken from Belgian websites:

Le Soir: Si, dans quinze jours ou trois semaines, on constate que l’épidémie retrouve sa courbe ascendante avec toutes ces souches mutantes, on peut imaginer qu’il faudra avancer l’heure du couvre-feu.

RTL info – Belgique: Leur voyage, prévu dans quinze jours, est annulé.

EDIT: see also the very good comment from @freddieknets: this reply above applies to native French speakers from Belgium. The original article quoted by OP refers to "Flamands", ie, to Flemish speakers using French as a 2nd language and transposing Flemish phrasings to French. In some extent, there has been an influence of Dutch/Flemish on Belgian French (ex: "une fois", "savoir/pouvoir" used indifferently), but that does not seem to be the case for this phrase.

I want to add to Greg’s answer, from my perspective (somebody from Flanders, the northern part of Belgium). We don’t speak French, but a regional variant of Dutch (Flemish). And we say "over veertien dagen" which would indeed translate to quatorze jours.

Why is this relevant? Because OP’s quote says:

Les Flamands pour exprimer deux ſémaines , diſent quatorze jours

Les Flamands, that’s us, the Northern Belgians. It should be noted that language is, and has always been, a sensitive issue in Belgium. Historically the only official language in Belgium was French (now it has three official languages), even though the people in the Flemish region have been speaking some form of Dutch since medieval ages. For a long time, this was just considered the language of the common folk, while French would be the choice of communication among the higher classes (even if their native tongue would still be Flemish). This only refers to the Northern parts of Belgium, in the South everybody has always been speaking French.

Hence, with the quote explicitly referencing les Flamands, I would assume that the difference that Laurent Chiflet observed in the 16th Century originated in Flemish people translating their tongue to French too literally, introducing a Germanism style error.