  The Navier-Stokes transport coefficients for a model of a confined
quasi-two-dimensional granular gas of smooth inelastic hard spheres are derived
from the Enskog kinetic equation. A normal solution to this kinetic equation is
obtained via the Chapman-Enskog method for states close to the local
homogeneous state. The analysis is performed to first order in spatial
gradients, allowing the identification of the Navier-Stokes transport
coefficients associated with the heat and momentum fluxes. The transport
coefficients are determined from the solution to a set of coupled linear
integral equations analogous to those for elastic collisions. These integral
equations are solved by using the leading terms in a Sonine polynomial
expansion. The results are particularized to the relevant state with stationary
temperature, where explicit expressions for the Navier-Stokes transport
coefficients are given in terms of the coefficient of restitution and the solid
volume fraction. The present work extends to moderate densities previous
results [Brey \emph{et al.} Phys. Rev. E \textbf{91}, 052201 (2015)] derived
for low-density granular gases.
