How do I convert a filter from Spatial to Fourier domain?
For a low-pass filter that is small compared to the image you can put your filter (unscaled) in the center of a square/rectangle the size and shape of your image (all the other pixels are 0), take the FFT and multiply the result  in the Fourier domain.

Since you are using a Gaussian filter you could do something more efficient than that.  The Gaussian filter is _ separable_ so you can actually create a 2D Gaussian filter by doing a 1D Gaussian filter in every row (column) in one dimension followed by a 1D Gaussian filter in every column (row) in the other dimension.

Second, the Fourier transform of a Gaussian is a Gaussian.  So we can work in one dimension at a time, we know what we're going to get, and multiplying a Gaussian in the Fourier domain is also separable since $e^{-(x^2+y^2)}=e^{-x^2}e^{-y^2}$.  The only question is what the standard deviation ($\sigma$) of the transformed Gaussian is going to be.  The derivation is worked out on Mathworld as:
$$
\mathcal{F}[e^{-ax^2}](k)=\sqrt{\frac{\pi}{a}}e^{-\pi^2k^2/a}.
$$