As Abtan requested, I'm converting my comments to an answer:
Suppose that $X$ is an $N$ (complex) dimensional complex manifold endowed with a Hermitean metric, or equivalently a Riemannian metric g satisfying $g(JX,JY)=g(X,Y)$, where $J$ is the complex structure. Let $*$ denote the $\mathbb{C}$-antilinear extension of the Hodge star operator to complex valued forms (some people -- including me -- prefer to write this as $\overline{*}$ as Spiro points out in the comments). Then as one finds on page 82 of Griffiths and Harris, $$*\Omega^{pq}\subset \Omega^{N-q,N-p}$$ where I'm following the notation in the question and writing $\Omega^{pq}$ for the space of $C^\infty$ forms of type $(p,q)$.