There is simple infinite family of simple solutions with $m=n=1$.
The system is linear in $k_i$, giving the simple solutions:
$$ k_1 = (m^2*m' - 1)*p^{(-t + 2)}/q^t \\ k_2 = (m'*n^2 - 1)*q^{(-t + 2)}/p^t \\ k_3 = (m*m'*n - 1)*p^{(-t + 1)}*q^(-t + 1) $$
Solution is $m'=p^t q^t+1$.