  We present the Laplace operator associated to a hyperbolic surface
$\Gamma\setminus\mathbb{H}$ and a unitary representation of the fundamental
group $\Gamma$, extending the previous definition for hyperbolic surfaces of
finite area to those of infinite area. We show that the resolvent of this
operator admits a meromorphic continuation to all of $\mathbb{C}$ by
constructing a parametrix for the Laplacian, following the approach by
Guillop\'e and Zworski. We use the construction to provide an optimal upper
bound for the counting function of the poles of the continued resolvent.
