The $p$ dimensional simples are the inductions of the 1-d reps of $C_q$. That is, the generators of order $p$ and $q$ act by the matrices

$A=\begin{bmatrix} 0 & 0 & \cdots &0 & 1\\ 1 & 0 & \cdots &0& 0\\ 0& 1 & \cdots &0& 0\\ \vdots&\vdots &\ddots&\vdots &\vdots\\ 0 & 0& \cdots &1& 0 \end{bmatrix}\,$ and $\,B=\begin{bmatrix} \zeta & 0 & \cdots &0& 0\\ 0& \zeta^b & \cdots &0& 0\\ \vdots&\vdots &\ddots&\vdots &\vdots\\ 0 & 0& \cdots &\zeta^{b^{p-1}}& 0\\ 0 & 0 & \cdots &0 & \zeta^{b^p}\\ \end{bmatrix}\,$ for $\zeta$ any $q$th root of unity, and $b$ an element with multiplicative order $p$ in $\mathbb{Z}/q\mathbb{Z}$. There are $d$ distinct ones, since these will be isomorphic if the same root of appears in the diagonal of $B$ in both.

The $p$ dimensional simples are the inductions of the 1-d reps of $C_q$. That is, the generators of order $p$ and $q$ act by the matrices

$A=\begin{bmatrix} 0 & 0 & \cdots &0 & 1\\ 1 & 0 & \cdots &0& 0\\ 0& 1 & \cdots &0& 0\\ \vdots&\vdots &\ddots&\vdots &\vdots\\ 0 & 0& \cdots &1& 0 \end{bmatrix}\,$ and $\,B=\begin{bmatrix} \zeta & 0 & \cdots &0& 0\\ 0& \zeta^b & \cdots &0& 0\\ \vdots&\vdots &\ddots&\vdots &\vdots\\ 0 & 0& \cdots &\zeta^{b^{p-2}}& 0\\ 0 & 0 & \cdots &0 & \zeta^{b^{p-1}}\\ \end{bmatrix}\,$
for $\zeta$ any $q$th root of unity, and $b$ an element with multiplicative order $p$ in $\mathbb{Z}/q\mathbb{Z}$. There are $d$ distinct ones, since these will be isomorphic if the same root of unity appears in the diagonal of $B$ in both.