  We present exact calculations of partition function $Z$ of the $q$-state
Potts model with next-nearest-neighbor spin-spin couplings, both for the
ferromagnetic and antiferromagnetic case, for arbitrary temperature, on
$n$-vertex strip graphs of width $L_y=2$ of the square lattice with free,
cyclic, and M\"obius longitudinal boundary conditions. The free energy is
calculated exactly for the infinite-length limit of these strip graphs and the
thermodynamics is discussed. Considering the full generalization to arbitrary
complex $q$ and temperature, we determine the singular locus ${\cal B}$ in the
corresponding ${\mathbb C}^2$ space, arising as the accumulation set of
partition function zeros as $n \to \infty$.
