About quadratic function and number theory

Question

If the vertex is above the straight line $$\{x\in\mathbb N, y = -4x+4\}$$ and the quadratic function with $(x-(n+1))^2-4n$ does not pass the (natural number, square number) point on the left side of the quadratic function, do there exist infinitely many cases where the quadratic function $$(x-(n-1))^2-4n+8$$ does not pass the (natural number, square number) point? Condition: $n \in \mathbb{N}$.