Given coprime $p,q$ and integer $t\geq3$. Consider the diophantine equations 
$$m'm^2=1+k_1p^{t-2}q^{t}$$ 
$$m'n^2=1+k_2p^tq^{t-2}$$
$$m'mn=1+k_3p^{t-1}q^{t-1}$$

Is there a solution where $m,n$ are coprime and $(mn,pq)=1$ and we get $m,n=O(\max({p^{\alpha(t-1)}},{q^{\alpha(t-1)}}))$ where $\alpha\in(1,2)$ holds?

From counting arguments it looks like there is room for $m,n=O(\max({p^{2(t-1)}},{q^{2(t-1)}}))$?