Given $P=xG$, prove in zero knowledge that I know $sqrt(x)$ Assuming that x has a sqrt. 

Given $P=xG$ is it possible to prove that I know the $sqrt(x)$ in zero knowledge?
My answer simply extends the comment by SEJPM.

Since the group has prime order (as you said in the comment), and since you assume that it is known that $x$ has a square root, you can simply prove knowledge of $x$ such that $xG = P$, using the standard Schnorr protocol for demonstrating knowledge of a discrete logarithm (see e.g. the wikipedia page, or my description here for a simplified security analysis of this protocol).

Now, since knowing $x$ is equivalent to knowing $\sqrt{x}$ in a group of prime order (each can be computed from the other in polynomial time), convincing the verifier that you know $x$ does also convince him that you know $\sqrt{x}$. Since the proof leaks nothing about $x$, it leaks nothing about $\sqrt{x}$.