I Wonder if طقس is a borrowing from Ancient Greek "Taxis", meaning "order, Disposition". The ط and ق representing Greek "t" and "k" seem to confirm this, because it is normal in arabic to represent plosive consonants of another language with arabic emphatics (as in إيطاليا for "Italy").
Interesting. It could be.
In fact, there is another word for the weather in Arabic but it is not widely used (maybe only in literature?) and that is النوء (An-naw') and from which the word for Meteorology is derived علم الأنواء
Away from this, the word طقس has also another usage which actually might confirm your theory. It is also used for "Ritual" and commonly in the plural form for Rituals طقوس.
In Greek orthodoxy, a rite/liturgy can also be designated as "Taxis" (in Roman catholicism you would have "ordo" as in "ordo missae"
Then I guess we made a discovery here :)
In fact, Greek has more influence on Arabic even before Islam, more than Latin did, because of the contact between (Eastern) Romans and Arabia back then and for centuries to come after.
One of the words mentioned in Quran which are said to be of Greek origin as well is the word سُنْدُس Sundus, meaning silk, or some type of silk. Not sure about it though, but the sound of "sundus" sounds parallel to how a Greek word would sound.