Let $P_0, P_1, P_2, ...$ be an enumeration of all logical formulas with one free variable $n$ (countable since they're finite strings of a countable alphabet). Let $Q(n) \iff \lnot P_n(n)$. For any $n$, $Q$ is not $P_n$ because they don't agree on $n$. But $Q$ is a logical formula so it should be in the enumeration...
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It's important to distinguish between strings representing valid formulas and the formulas themselves.
See the following:
This will come down to the following: what is the precise definition of "logical formula?" The point is that although the $Q$ you've described is definitely "meaningful," it is of greater complexity in a sense than any of the $P_i$s. When we pin down a precise notion of "logical formula" and an enumeration of such, e.g. "first-order formula in the language of arithmetic" ordered lexicographically via some reasonable ordering on the symbols involved, the corresponding $Q$ will turn out to not be expressible by a formula in that particular sense.
For a concrete example of how this plays out, see e.g. Tarski's undefinability theorem. And compare this to similar "paradoxes of expression:" Berry's paradox, Richard's paradox, and Grelling's paradox.