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 in Bernhard Baumgartner, An Inequality for the trace of matrix products, using absolute values.
Another generalization is very similar to what you wrote and reads
$$ \parallel|AB|\parallel \, \le\, \parallel |A|^p\parallel^{1/p} \,\, \parallel|B|^q \parallel^{1/q} $$ where $|M|:=(M^\dagger M)^\frac12$ and it holds whenever $ \parallel \cdot \parallel$ is a unitarily invariant norm. You can find a proof in the book of Bhatia Matrix Analysis.
