Generalized Fibonacci and Lucas Numbers of the form $5x^2$

Let $(U_n(P,Q)$ and $(V_n(P,Q)$ denote the generalized Fibonacci and Lucas sequence, respectively. In this study, we assume that $Q=1.$ We determine all indices $n$ such that $U_n=5\mathrm{\square }$ and $U_n=5U_m\mathrm{\square }$ under some assumptions on $P\mathrm{.}$ We show that the equation $V_n=5\mathrm{\square }$ has the solution only if $n=1$ for the case when $$ is odd. Moreover, we show that the equation $V_n=5V_m\mathrm{\square }$ has no solutions.