  The K-moduli theory provides a different compactification of moduli spaces of
curves. As a general genus six curve can be canonically embedded into the
smooth quintic del Pezzo surface, we study in this paper the K-moduli spaces
$\overline{M}^K(c)$ of the quintic log Fano pairs. We classify the strata of
genus six curves $C$ appearing in the K-moduli by explicitly describing the
wall-crossing structure. The K-moduli spaces interpolate between two birational
moduli spaces constructed by GIT and moduli of K3 surfaces via Hodge theory.
