Are there any two positive integers such that the square root of the sum of their squares give a perfect square. Are there two positive integers $$ \text{{ (x,y) | $ \sqrt{x^2 + y^2} = z $  }}$$
where $ z$ is a perfect square.

If you're looking for integers such that $\sqrt{x^2+y^2}=z$, then there's plenty, for example, $3,4,5$ or $12,5,13$.
If you're looking for integers such that $\sqrt{a^2+b^2}=c^2$, then take any solutions to the first equation and multiply everything by $z$, e.g. take $3,4,5$ and multiply by $5$.
Then, since
$$\sqrt{x^2+y^2}=z,$$
it follows that
$$\sqrt{(xz)^2+(yz)^2}=\sqrt{z^2(x^2+y^2)}$$
$$=z\sqrt{x^2+y^2}=z^2$$
For example, $15^2+20^2=25^2$.
