Aliasing due to the Convolution of Gaussian Functions
Short answer: For the convolution of two Gaussian sequences there will be aliasing.

If the sequence produced as the result of linear convolution has infinite domain of support, then there will always be aliasing in the Fourier domain implementation of the linear convolution using the circular convolution implied by the DFT (discrete Fouerier transform) which is used to represent the theoretical DTFT that provides the theorem in your question.

Therefore, if you want an alias free implementation of the theorem in your question, then the resulting signal (result of the true linear convolution) must have a finite length.

That being said, for many practical cases, the amount of distortion due to aliasing will be small, so small to be lost behind any ADC step size or even smaller than represantable by the floating point number system that's used to implement the algorithm. In such cases it can practically be considered to be a sufficient realization of the true lienar convoluion.