Get points of an Elliptic Curve defined over a Finite Field on Twisted Edwards Extended Coordinates
You will have a weistrass equation of the form:

$Y^2 = aX^3..$

Where x is the only unknown variable on the right hand side. Insert a random X value in to the right hand side. Then evaluate the right hand side. This would need to be done modulo whatever prime you are using.

You will end up with an equation of the form:

$Y^2 = K$ where K is your evaluated right hand side. 

This number K is either a Quadratic Residue or it is not. You can check this using the Legendre symbol.

If it is, then you can perform the sqrt of K modulo your prime to get the value for $+Y, -Y$ 

How you do the sqrt depends on what algorithm you decide to use, you can use Tonelli-Shanks which is a more generic square root algorithm