Does the independence of the axiom of choice imply Gödel's incompleteness theorem?

The answer depends on what you mean by "the incompleteness theorems". If all you mean is "$ZF$ is incomplete", then yes, the independence of $AC$ is enough to prove that $ZF$ is incomplete (though it's worth remembering that the consistency of $\neg AC$ was proved much later than Gödel's incompleteness theorems).

However, Gödel actually proved statements stronger than just "$ZF$ is incomplete". For example, the first incompleteness theorem tells you that (as long as $ZF$ is consistent) not only is $ZF$ incomplete, but you can't make it complete by adding any computably enumerable list of axioms to it. The second incompleteness theorem tells you specifically that (again, assuming that $ZF$ is consistent) one of the things $ZF$ can't prove is $Con(ZF)$. This is important because there are statements of interest in set theory (such as the consistency of large cardinals) that do imply $Con(ZF)$, and hence we know that $ZF$ can't prove that these statements are true (but remember, knowing that you can't prove $\sigma$ isn't the same thing as proving $\neg\sigma$!).

With ZF and AC, it is the case that one particular set of axioms (such as ZF) is incomplete (since ZF implies neither AC nor $\lnot$ AC).

Gödel incompleteness theorem states that every [computable and consistent] set of axioms [strong enough to model arithmetic] is incomplete. So you cannot add a [computable and consistent with ZF] set of axioms to ZF to make it complete.

As the other answers have said, the independence of $\mathsf{AC}$ over $\mathsf{ZF}$ merely suffices to establish a specific case of the incompleteness theorem: that $\mathsf{ZF}$ is not a complete theory. (All that assumes that $\mathsf{ZF}$ is consistent of course!)

However, there's also an important positive aspect here. Gödel's theorem gives a way to assign to any "appropriate" theory $T$ a sentence $\sigma_T$ which is independent of $T$. But this $\sigma_T$ isn't a very interesting sentence on its own - there's no obvious reason to care about it except because its analysis gives us the incompleteness of $T$. By contrast, Cohen and Gödel's work on $\mathsf{AC}$ shows that there is an interesting sentence which is independent of $\mathsf{ZF}$. That's the sort of thing which the incompleteness theorem can't give us on its own (unsurprisingly, since it's an informal statement): a priori there's no reason we couldn't have some "appropriate" theory $T$ that, while incomplete per Gödel, does decide every sentence that actually arises in non-logic-focused mathematics. (E.g. $\mathsf{ZFC+V=L}$ seems to come pretty close to this.)

There is a general attitude - to be fair, I don't know how general, but at least I'm an ardent believer - of "Gödelian optimism" (or "Gödelian pessimism," depending who you talk to): that in fact every "appropriate" theory will have some natural sentence independent of it. The incompleteness theorem only sets the stage for this, it doesn't actually get us all the way there. Gödel/Cohen demonstrate this convincingly for the particular case of $\mathsf{ZF}$ (and Cohen's method of forcing quickly demonstrates the same for many extensions of $\mathsf{ZF}$).

(FWIW, one weak point of evidence in favor of Gödelian optimism is that as a corollary of the incompleteness theorem the set of sentences independent of an "appropriate" theory $T$ is never computable. So there won't ever be a "single reason" that things are independent of $T$. But in my opinion this is still very weak evidence.)