  In this article, we consider the quasi-linear stochastic wave and heat
equations on the real line and with an additive Gaussian noise which is white
in time and behaves in space like a fractional Brownian motion with Hurst index
$H\in (0,1)$. The drift term is assumed to be globally Lipschitz. We prove that
the solution of each of the above equations is continuous in terms of the index
$H$, with respect to the convergence in law in the space of continuous
functions.
