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.