  In this paper, we study a certain deformation $D$ of the Virasoro algebra
that was introduced and called $q$-Virasoro algebra by Nigro,in the context of
vertex algebras. Among the main results, we prove that for any complex number
$\ell$, the category of restricted $D$-modules of level $\ell$ is canonically
isomorphic to the category of quasi modules for a certain vertex algebra of
affine type. We also prove that the category of restricted $D$-modules of level
$\ell$ is canonically isomorphic to the category of $\mathbb{Z}$-equivariant
$\phi$-coordinated quasi modules for the same vertex algebra. In the process,
we introduce and employ a certain infinite dimensional Lie algebra which is
defined in terms of generators and relations and then identified explicitly
with a subalgebra of $\mathfrak{gl}_{\infty}$.
