The frequency spectrum of a static non-linearity driven by colored noise
If your nonlinearity can be expressed as a polynomial (i.e., in terms of addition and multiplication), you can make use of:


The linearity of the Fourier transform, i.e., if $f$ and $g$ are (benign) functions, $a$ and $b$ are numbers and $ℱ$ denotes the Fourier transform, then:

$$ℱ(a·f+b·g) = a·ℱ(f) + b·ℱ(g)$$
The convolution theorem, which states that multiplication in the time domain corresponds to convolution in the frequency domain (and vice versa):

$$ℱ(f·g) = ℱ(f) ∗ ℱ(g)$$


So, in your specific example, the spectrum of $y$ is the spectrum of $v$ convoluted with itself:

$$ℱ(y) = ℱ(v·v) = ℱ(v) ∗ ℱ(v)$$