There are several definitive results regarding perfect powers in the Pell numbers — e.g., the only perfect power is $P_7=169=13^2$.
On the other hand, when it comes to powerful numbers, I've only found “non-effective” results [sorry, I don't know a better word for it!], such as those by Ribenboim and Yabuta‘s that the ABC conjecture implies that there are finitely many powerful numbers in each Lucas sequence.
Are there any results which are more definitive, e.g. “if $d$ has property X, then $v$ in $u^2-dv^2=1$ is powerful if and only if Y”?