Estimating the Nyquist rate when simulating data
You must know the method of computing Q(t), and if Q(t) = Q[f(t)], meaning that Q is a function of another function, like the output of a filter. Then, knowing the autocorrelation or power spectrum of f(t), one can use various methods to find the power spectrum of Q. If the function Q is linear with respect to f(t), then the solutions are simple.

However, it seems like you should clear up your needs. An FFT of the signal is simply its spectrum, not is power spectrum or its power spectral density. They are slightly different.

To answer your last question, it seems like you are formulating this in a probability/statistics sense when you say independent? If they are truly independent (i.e. one value of Q gives absolutely no clue about the next value), then your power spectral density represents white noise and it is flat (a constant). Again, it would be best if you clarified your definition of "independent". For example, if Q is a relatively smooth function of time, then it is not statistically "independent" as you say, and to find the spectrum or power spectrum density, you should analytically look at other functions that Q depends on first.