  Given $k\ge 2$ and two $k$-graphs ($k$-uniform hypergraphs) $F$ and $H$, an
\emph{$F$-factor} in $H$ is a set of vertex disjoint copies of $F$ that
together covers the vertex set of $H$. Lenz and Mubayi studied the $F$-factor
problems in quasi-random $k$-graphs with minimum degree $\Omega(n^{k-1})$. In
particular, they constructed a sequence of $1/8$-dense quasi-random $3$-graphs
$H(n)$ with minimum degree $\Omega(n^2)$ and minimum codegree $\Omega(n)$ but
with no $K_{2,2,2}$-factor. We prove that if $p>1/8$ and $F$ is a $3$-partite
$3$-graph with $f$ vertices, then for sufficiently large $n$, all $p$-dense
quasi-random $3$-graphs of order $n$ with minimum codegree $\Omega(n)$ and
$f\mid n$ have $F$-factors. That is, $1/8$ is the density threshold for
ensuring all $3$-partite $3$-graphs $F$-factors in quasi-random $3$-graphs
given a minimum codegree condition $\Omega(n)$. Moreover, we show that one can
not replace the minimum codegree condition by a minimum vertex degree
condition. In fact, we find that for any $p\in(0,1)$ and $n\ge n_0$, there
exist $p$-dense quasi-random $3$-graphs of order $n$ with minimum degree
$\Omega (n^2)$ having no $K_{2,2,2}$-factor. In particular, we study the
optimal density threshold of $F$-factors for each $3$-partite $3$-graph $F$ in
quasi-random $3$-graphs given a minimum codegree condition $\Omega(n)$.
