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I have a phrase, let's say A = (∀x)(∀y)F(G(x, y), x)

Now, how do I convert ¬A into CNF? I believe applying ¬ to A would change the global quantifiers to existential ones, so I come to this as an answer:

¬F(G(E(x), E(y)), E(x)))

I can't, however, find anything about this online, if someone could tell me the correct one I'd be most grateful.

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It depends on what you are trying to do. If you want to prove that $A$ is valid by proving that $\neg A$ is unsatisfiable, the change to existential quantifiers is just what you want and there is no need to eliminate them.

On the other hand, if you insisted on eliminating the quantifiers, in this case you'd need two Skolem constants, because the existential quantifiers are not nested inside universal quantifiers. Hence, you'd get somethings like $\neg F(G(a,b),a)$, where $a$ and $b$ are "fresh" (new) constant symbols.

Regardless, if $F$ is a relation symbol and $G$ is a function symbol, the expression $\neg F(G(x,y),x)$ is just the negation of an atom. Hence it is already in CNF.

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    F and G are just predicates. I was just asked to find the CNF of ¬A, it's not part of an exercise, just reading up for my finals and this is an unanswered question in the class notes. – 2017-01-12
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    @JustACuriousMind OK, I've updated my answer accordingly. Syntactically, $G$ must be a *function* symbol. – 2017-01-12