No, “*Part le train pour Paris de ce quai ?” is not correct in mainstream French nor in any dialect I’ve heard of. Verb-subject inversion to form a question is only possible when the subject is a personal pronoun (je, tu, il, elle, on, nous, vous, ils, elles) or the pronoun ce.

In practice, it would probably be understood in writing provided that the question mark is present, but it would require closer reading and be perceived as an obvious mistake. In spoken French, a question is indicated by the rising tone at the end of the sentence, so word order doesn’t matter (this is what allows “Le train pour Paris part de ce quai ?” to be a question in colloquial French), but once again the sentence would be perceived as incorrect and might be difficult to understand (e.g. it might be initially understood as “Par le train de Paris …”).

In formal French, to form a question where the subject is a pronoun, you put the subject after the verb, with a hyphen in between.

Part-il de ce quai ?
Sont-elles climatisées ?
Viendrez-vous ce soir ?
Quand part-il ?   (It’s the same thing with or without an interrogative particle.)
Est-il parti de ce quai ?   (With a compound verb, the subject goes after the first auxiliary.)

If the subject is anything else, you put the real subject before the verb, and add a pronoun after the verb to serve as a second grammatical subject.

Le train part-il de ce quai ?
Les voitures sont-elles climatisées ?
Ton mari et toi viendrez-vous ce soir ?
Quand le train part-il ?

In the medium-formality construct est-ce que, the pronoun ce is an inverted subject. In the rest of the sentence (“Est-ce que le train pour Paris part de ce quai ?”), which is formally a subordinate clause introduced by que, the subject comes before the verb. The pronoun ce used as a subject of être can be inverted as well.

Ce que tu m’as dit est-il vrai ? → Est-ce vrai ?

(Some dialectal forms allow “C’est-il vrai ?”, but this is not mainstream contemporary French.)