Square Values of a Quadratic

The quadratic $f(x) = x^2 + bx + c$ passes through $(1, 13)$, and $y = f\left(x - \tfrac12\right)$ is even.

1. Find $f(x)$. 2. Are there points on $y = f(x)$ whose $x$-coordinate is a positive integer and whose $y$-coordinate is a perfect square? Find all such points or prove none exist.