  We investigate the thermal conductivity ($\kappa$) of the quark matter at
finite quark chemical potential $(\mu)$ and temperature $(T)$, employing the
Green-Kubo formula, for the SU(2) light-flavor sector with the finite
current-quark mass $m=5$ MeV. As a theoretical framework, we construct an
effective thermodynamic potential from the $(\mu,T)$-modified liquid-instanton
model (mLIM). Note that all the relevant model parameters are designated as
functions of $T$, using the trivial-holonomy caloron solution. By solving the
self-consistent equation of mLIM, we acquire the constituent-quark mass $M_0$
as a function of $T$ and $\mu$, satisfying the universal-class patterns of the
chiral phase transition. From the numerical results for $\kappa$, we observe
that there emerges a peak at $\mu\approx200$ MeV for the low-$T$ region, i.e.
$T\lesssim100$ MeV. As $T$ increase over $T\approx100$ MeV, the curve for
$\kappa$ is almost saturated as a function of $T$ in the order of
$\sim10^{-1}\,\mathrm{GeV}^2$, and grows with respect to $\mu$ smoothly. At the
normal nuclear-matter density $\rho_0=0.17\,\mathrm{fm}^{-3}$, $\kappa$ shows
its maximum $6.22\,\mathrm{GeV}^2$ at $T\approx10$ MeV, then decreases
exponentially down to $\kappa\approx0.2\,\mathrm{GeV}^2$. We also compute the
ratio of $\kappa$ and the entropy density, i.e. $\kappa/s$ as a function of
$(\mu,T)$ which is a monotonically decreasing function for a wide range of $T$,
then approaches a lower bound at very high $T$:
$\kappa/s_\mathrm{min}\gtrsim0.3\,\mathrm{GeV}^{-1}$ in the vicinity of
$\mu=0$.
