Can we get the same result from between partial FFT? I'm trying to implement fft and gaussian low pass filter multiplication to 16x16 image.

As I know fft can be separable. What if there are 16x16 one image, then is this the same ? between 4x4 part of the whole 16x16 fft result and  just only 4x4 fft result?

If they are same then I think that the latter method is more resource efficienciable.
I don't think "separable" means what you think it means – it means that you can implement the 2 DFT as the concatenation of a $N_{rows}$ amount of $N_{columns}$ long DFTs followed by $N_{columns}$ DFTs of length $N_{rows}$.

So, no, your approach is not correct.