No.
Gandhara finds its name in way older documents such as Mahabharata. The people of Gandhara were called Gandharva. According to Mahabharata, these were people skilled in archery, wars, and also fine arts.
I am not sure about the history, but I am sure the name doesn’t come from Alexandria.
This sort of thing happened a lot with the Old Testament (despite being written down). I would be shocked if it didn't also happen with Mahabharata, which is famous for not having been written down (much).
Priests were highly trained in recitation and memorization and the oral texts were consistent across vast regions.
My source is vague memory but I think it was from JAB van Buitenen speaking about oral transmission and accuracy in the intros for either “The Bhagavadgita in the Mahabharata” or “The Mahabharata; Book of the Beginning”.