Fix a prime $p$.
How many Sylow $p$-groups are there in $GL_3(\Bbb F_p)$?
Let $s_p$ be the number of Sylow $p$-groups in $GL_3(\Bbb F_p)$. By the Sylow theorems, $s_p\equiv 1\mod p$ and if $p^e$ is the maximal prime power of the order of $GL_3(\Bbb F_p)$, then $s_p$ divides $\frac{\left|GL_3(\Bbb F_p) \right|}{p^e}$. Unfortunately I don't even really know how to figure out the cardinality of that group.
I assume you use column vectors.
One particular Sylow $p$-subgroup is $$ L = \left\{ \begin{bmatrix} 1 & 0 & 0\\ a & 1 & 0\\ c & b & 1\\ \end{bmatrix} : a, b, c \in \Bbb F_p \right\}. $$ You may try and compute its normalizer. The index will give you the number of Sylow $p$-subgroups.
Alternatively, you may note that $x \in L$ iff $x - 1$ maps the underlying vector space to $\langle e_{2}, e_{3} \rangle$, $\langle e_{2}, e_{3} \rangle$ to $\langle e_{3} \rangle$ and $\langle e_{3} \rangle$ to $\{ 0 \}$, where $e_{1}, e_{2}, e_{3}$ is the standard basis. So there are as many Sylow $p$-subgroups as the number of pairs $U_{2} \supseteq U_{1}$ of subspaces, with $\dim(U_{i}) = i$. You will obtain the same number as with the previous method.