This question (and its comments) put forward the following information:
Theorem: Let $\mathbb{F}_q$ be a finite field with odd $q\neq 9$ elements and $a$ be a generator of the cyclic group $\mathbb{F}_q^{\times}$. Show that $\mathrm{SL}(2,q)$ is generated by $$A=\begin{pmatrix} 1 & 1 \\ 0 & 1 \\ \end{pmatrix}$$ and $$B=\begin{pmatrix} 1 & 0 \\ a & 1 \\ \end{pmatrix}.$$
and a proof can be found in Suzuki's, "Group Theory I".
I would like to see a proof soon or, at any rate, have access to one; but I can't get a hold of Suzuki's book. It isn't at my university library. I have a physical copy of my own. It's with a relative in England, though, and I'm currently in Scotland. Hence the alternative-proof tag: I want an alternative reference, please.
If the proof is simple, please share it, although I haven't warranted such an answer with enough context.
Some thoughts:
The way I'd try to prove it is to look at the forms of the conjugacy classes of ${\rm SL}(2,q)$ then attempt to construct their representatives from $A$ and $B$, to see what ideas come to mind.