How can I put this sentence into prenex form I have this sentence, and want to put it in Prenex form: $$ \lnot(( \exists y)( \forall z) Q(y,z)) $$ Is it this?: $$ ( \forall y)( \exists z) \lnot Q(y,z) $$

Yes ... basically the relevant rule is:

> When you push/pull a negation sign past an _adjacent_ quantifier, the quantifier flips into its dual.

So pushing inwards, $\
eg\forall x\varphi$ is equivalent to $\exists x\
eg\varphi$. Pulling outwards $\forall x\
eg\varphi$ is equivalent to $\
eg\exists x\varphi$.

That's because $\forall$ and $\
eg\exists\
eg$ come to the same: so $\
eg\forall x\varphi$ is equivalent to $\
eg\
eg\exists x\
eg\varphi$ is equivalent to $\exists x\
eg\varphi$. Similarly for $\exists$ and $\
eg\forall\
eg$, and so on (all classically, of course).

So $\lnot\exists y\forall z Qyz$ is equivalent to $\forall y\
eg\forall z Qyz$ is equivalent to $\forall y\exists z\
eg Qyz$

[Minor point. Your notation is unnecessarily over-bracketed according to most modern versions of FOL syntax.]