Given coprime $p,q=\Theta(\ell)$ and integer $t\geq3$ is there coprime $m,n=\Theta(\ell^{t-1})$ with $(mn,pq)=1$ such that there are $\alpha_i\in\Bbb Z$ and $m'\in\Bbb Z$ with
$$\alpha_1p^tq^t+1=m'm^2$$
$$\alpha_2p^tq^t+1=m'mn$$
$$\alpha_3p^tq^t+1=m'n^2$$
where $m,n$ where $0<m'<p^{t}q^t$ holds?

There seems to be room for such $m',m,n,\alpha_i$.

$\alpha_i=\Theta(\ell^{2t})$ holds.

The expected intersection of choices is $$\frac{\mbox{# choices of }\alpha_i\times\mbox{# choices of }m',m\mbox{ and }n}{(\mbox{# of choices of resiudes modulo }p^tq^t)^3}\approx\frac{\ell^{6t}\times \ell^{2t}\ell^{t-1}\ell^{t-1}}{\ell^{6t}}=\ell^{4t-2}\gg1.$$