There are (at least two) "generalizations" of Hölder inequality to the non-commutative case. One is the so called tracial matrix Hölder inequality:
$$ |\langle A, B \rangle_{HS} |= |\mathrm{Tr} (A^\dagger B) | \le || A||_p \,\, || B||_q $$
where $|| A||_p$ is the Schatten $p$-norm and $1/p+1/q=1$. You can find a proof here.
Another generalization is very similar to what you wrote and reads
$$ \parallel \, |AB|\, \parallel \, \le\, \parallel \, |A|^p\, \parallel^{1/p} \,\, \parallel \, |B|^q\, \parallel^{1/q} $$
and it holds whenever $ \parallel \cdot \parallel$ is a unitarily invariant norm. You can find a proof in the book of Bhatia Matrix Analysis.