I suspect the Granville-Langevin conjecture implies this.

Let $F(x,y)=0$ be a curve with infinitely many integral points $(F_n,F_{n+1})$.

Is it true that either $x^2+x y - y^2 -1$ or $x^2+ x y -y ^2 +1$ divides $F(x,y)$?

Probably this can be generalized to integral points $(F_{f(n)},F_{g(n)})$.