Is there a quick method of listing certain elements of a cyclic group?
Since this is a field, you know that the polynomial $x^2-2=0$ may have up to 2 roots, no more. You could use the quadratic formula (since the polynomial ring over a field is an integral domain) but this would require the extraction of a square root. But it is certainly more efficient than checking all elements.

Also if if $a$ in $x^2-a$ is a square then you can factor $(x^2-a)=(x-b)(x+b)$ where $b^2=a.$