Steinberg's "Lecture Notes on Chevalley Groups" the Corollary of Lemma 54 on page 132. There is possibly a more modern reference.

EDIT: Sorry the reference to Steinberg is not sufficient as he does not treat arbitrary finite reductive groups. This follows from a theorem of Rosenlicht about the fixed points of a unipotent group under a Frobenius endomorphism. If $\mathbf{G}$ is a connected reductive algebraic group with Frobenius endomorphism $F : \mathbf{G} \to \mathbf{G}$ and $\mathbf{U} \leqslant \mathbf{G}$ is an $F$-stable unipotent subgroup then the finite group $\mathbf{U}^F$ has order $q^{\dim\mathbf{U}}$, (see Geck's book "An Introduction To Algebraic Geometry and Algebraic Groups" - Theorem 4.2.4). Now take $\mathbf{G} = \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ and $F$ to be the map given by $F(x_{ij}) = (x_{ij}^q)$ where $q = p^a$ for some integer $a$ then $\mathbf{G}^F = \mathrm{GL}_n(q)$. The subgroup $\mathbf{B} \leqslant \mathbf{G}$ of upper triangular matrices is an $F$-stable Borel subgroup of $\mathbf{G}$ and its unipotent radical is the subgroup of all upper uni-triangular matrices. This has maximal dimension amongst all unipotent subgroups of $\mathbf{G}$, which can be seen in the following way. Assume $\mathbf{V} \leqslant \mathbf{G}$ is a unipotent subgroup of $\mathbf{G}$ then as $\mathbf{V}$ is solvable it is contained in a Borel subgroup $\mathbf{B}'$ of $\mathbf{G}$. As $\mathbf{V}$ is a unipotent subgroup of $\mathbf{B}'$ it is contained in the unipotent radical of $\mathbf{B}'$. As all Borel subgroups of $\mathbf{G}$ are conjugate we have the unipotent radicals of $\mathbf{B}$ and $\mathbf{B}'$ have the same dimension, hence we have $\dim\mathbf{V} \leqslant \dim \mathbf{U}$. Now applying Rosenlicht's theorem we have $\mathbf{U}^F$ is a Sylow $p$-subgroup of $\mathbf{G}^F$, which is precisely the subgroup in question. This approach has the advantage that it works for all finite reductive groups.