On the groupprops wiki for $SL_2(\mathbb F_q)$ it says there are 4 conjugacy classes in the case of a Jordan block of size 2.
Aren't the possible forms just $\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, \begin{pmatrix} -1 & 1 \\ 0 & -1 \end{pmatrix}$? What else is there?
These are the only possible Jordan forms. But it is possible for two matrices to have the same Jordan form, but not be conjugate under $\mathrm{SL}_{2}(\mathbb{F}_{q})$ (though according to the table you link to, they will still be conjugate under $\mathrm{GL}_{2}(\mathbb{F}_{q})$).
For example, $\begin{bmatrix} 1&1\\0&1 \end{bmatrix}$ and $\begin{bmatrix} 1&-1\\0&1 \end{bmatrix}$ have the same Jordan form, but are not equivalent under $\mathrm{SL}_{2}(\mathbb{F}_{q})$.
The two representatives you are missing are $\begin{bmatrix} 1&-1\\0&1 \end{bmatrix}$ and $\begin{bmatrix} -1&-1\\0&-1 \end{bmatrix}$
Note that this all assumes that $q$ is odd.