  Let $p$ be a prime and $q = p^k$. A subset $\mathcal{F} \subset
\operatorname{\Gamma L}_{2}(q)$ is intersecting if any two semilinear
transformations in $\mathcal{F}$ agree on some non-zero vector in
$\mathbb{F}_q^2$. We show that any intersecting set of $\operatorname{\Gamma
L}_{2}(q)$ is of size at most that of a stabilizer of a non-zero vector, and we
characterize the intersecting sets of this size. Our proof relies on finding a
subgraph which is a lexicographic product in the derangement graph of
$\operatorname{\Gamma L}_{2}(q)$ in its action on non-zero vectors of
$\mathbb{F}_q^2$. This method is also applied to give a new proof that the only
maximal intersecting sets of $\operatorname{GL}_{2}(q)$ are the maximum
intersecting sets.
