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 $(F\_{f(n)},F\_{g(n)})$.



Curves with infinitely many integral points consecutive Fibonacci numbers
You don't need any conjectures for this. If two curves have infinitely many points in common then they have a common component (this is a basic fact of the Zariski topology). As $x^2+xy-y^2-1$ and $x^2+xy-y^2+1$ are irreducible, one of them must divide $F$.


