Special Full body washing "Ghusl" in the Quran? No.

Rilum

443 O you who believe approach not the Prayer when you are intoxicated until you understand what you say; nor when you are unclean (save passing by upon the path), until you wash. And if you were ill, or on a journey, or one of you comes from the privy, or you have lain with women, then find not water resort to clean soil, and wipe your faces and your hands; God is pardoning and forgiving.

One should not approach prayer while being intoxicated, or in a state of impurity, until one is washed. If you understand the verb wash as "Ghusl", then you would have to do the whole body wash even if you go to the toilet.
God does not make an extra rule that only if one has had sexual intercourse one should then do "Ghusl".

56
O you who believe when you rise up for the Prayer wash your faces, and your hands to the elbows, and wipe your heads, and your feet to the ankles. And if you are unclean, purify yourselves. And if you were ill, or on a journey, or one of you came from the privy, or you have lain with women, then find not water resort to clean soil, and wipe your faces and your hands with it. God wishes not to place any distress upon you; but He wishes to purify you and to complete His favour upon you, that you might be grateful.

The word "wash" in 443 does not refer to an invented Ghusl command, but to the ablution in 56, in which certain parts of the body are washed and stroked.
So there is no special full body wash called Ghusl in the Quran that you only have to do when you have sexual contact with women.

Aladin_Azra

We should first re-think about translations. There're many issues, but I'll start with one in 56.

إِذَا قُمْتُمْ إِلَى الصَّلَاةِ

This is translated in a provided translation as "when you rise up for the Prayer".

Time adverb idhaa is translated same as idh, which is not correct. Idh means "when", but "idhaa" means "after", both are time adverbs but slightly different.

Verb with a pronoun qum-tum is in perfect tense describing action of standing, not moving/changing position. Just by changing time adverb the whole meaning could be changed with a little change of other words. I'll take this as "you stood", not "stand" nor "stand up" nor "rise up".

Preposition ilaa means to/towards. Translated as "for" is wrong.

A potential definitive feminine noun eS-Salaat can name only a potential of an action, not an action itself. Even if the root S-l-w means "to pray" (and it doesn't) then this noun should describe something that potentially can be praying, not praying itself.

This could be translated as follows "After you stood to/towards Salaat". Just time adverb can show that this has nothing with a prayer of Persian buddists called "nimas" and that an imperative igsuloo and all other words should mean something else then we were told. If you accept that idhaa is same as idh I think it would be very hard for you to find the answer you're searching for.

I'm glad you're questioning this topic and, if I can, I should advice to you to start from the noun used in 6936. Also verb and participle in form VIII were used, and this form is a reflexive of main form, so they can also help with understanding that imperative verb.