Is this formula in normal form prenex $(\forall x)(\exists y)(P(x) \land (\exists z)Q(y, z))$? In my course notes, this $(\forall x)(\exists)z(P(x) \land (\exists z)Q(y, z))$ formula is said to be in `normal form prenex`. But, shouldn't al cuantors be in the back of the formula? I mean, I would expect this formula to be in the normal form prenex: $(\forall x)(\exists y)(\exists z)(P(x) \land Q(y, z))$

Yes prenex normal form must have all quantifiers at the front so what you said is correct.