On squared Integral points of a Parabola

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I am puzzling about one pretty weird question which I cannot find in Internet.

Let $y=ax^2+bx+c$ a parabola where of course $x \in\mathbb R$ and $a,b,c \neq 0$

It's easy to understand that for any $x \in\mathbb N$ the values of $y$ are integer numbers too.

But would it be possible to “know” which $x \in\mathbb N$ give $y \in\mathbb N$ values (so we have integral points) AND $y$ is a square ? (So $\sqrt{y} \in\mathbb N$) ?

Sorry for the naive approach but this is my first post here and my first attempt to use MathJax…

Thanks, Davinator