If I put "quite" after "a" like "She's a quite girl", does the meaning stay the same?
Unfortunately, that would be grammatically incorrect and would not have any meaning. Quite is an adverb meaning it needs to come before a verb, adjective or article+noun. As it is not an adjective it cannot simply be placed with a noun on its own. In this context, it is being used to emphasize that someone (the girl) is impressive, unusual or interesting. Ex: That's quite an outfit! Ex: You're quite the artist!