Yes, ${\rm PGL}(2,q^2)$ is a subgroup of ${\rm PGL}(4,q)$, but I would guess that that is an exception, and in general there is no such embedding.

${\rm GL}(4,q)$ contains the subgroup that I denote by ${\rm CO}^-(4,q)$, which is the conformal orthogonal group of minus-type (and equal to the normalizer in ${\rm GL}(4,q)$ of ${\rm GO}^-(4,q)$).

The projective image ${\rm PCO}^-(4,q)$, which is of course a subgroup of ${\rm PGL}(4,q)$, happens to be isomorphic to ${\rm P \Gamma L}(2,q^2)$, which contains ${\rm PGL}(2,q^2)$ as a subgroup of index $2$.

So this is a separate embedding, and is not related to the semiliear embedding ${\rm GL}(2,q^2) \to {\rm GL}(4,q)$.