Can realizations of a filtered Gaussian white noise process be represented as a Fourier transform?
I don't think it makes sense to talk about the Fourier transform of a (realization of a) random process, because it generally doesn't exist if no further restrictions are imposed. This problem is also discussed here.

What can be done is analyze the statistics of the discrete Fourier transform (DFT) and the discrete-time Fourier transform (DTFT) of a windowed version of a discrete-time stationary random process. This has been done in this paper.