Let
$\mathbb{F}_{q}^{n}$
be a vector space of dimension
$n$
over the finite field
$\mathbb{F}_{q}$
. A
$q$
-analog of a Steiner system (also known as a
$q$
-Steiner system), denoted
${\mathcal{S}}_{q}(t,\!k,\!n)$
, is a set
${\mathcal{S}}$
of
$k$
-dimensional subspaces of
$\mathbb{F}_{q}^{n}$
such that each
$t$
-dimensional subspace of
$\mathbb{F}_{q}^{n}$
is contained in exactly one element of
${\mathcal{S}}$
. Presently,
$q$
-Steiner systems are known only for
$t\,=\,1\!$
, and in the trivial cases
$t\,=\,k$
and
$k\,=\,n$
. In this paper, the first nontrivial
$q$
-Steiner systems with
$t\,\geqslant \,2$
are constructed. Specifically, several nonisomorphic
$q$
-Steiner systems
${\mathcal{S}}_{2}(2,3,13)$
are found by requiring that their automorphism groups contain the normalizer of a Singer subgroup of
$\text{GL}(13,2)$
. This approach leads to an instance of the exact cover problem, which turns out to have many solutions.