Can low-quality images be used in FreeSurfer?

If the images are, indeed, low quality, would that mean that they should not be processed by freesurfer at all or will it just take longer to get the same quality image?

You won’t get better quality images by waiting longer. The issue with low-quality images is that they may produce bad first-pass estimates of the gray-/white-matter boundary surface, producing a surface with handles that need to be removed (topological correction) to get a surface that’s topologically equivalent to a sphere. Some handles will appear in the best of cases, but low-quality images often result in many handles, which makes the problem much harder, and a good solution may not be converged on.

I don’t think it’s a bad idea to try to run FreeSurfer, but some images are unrecoverably poor. If you can’t see where you would place the boundary surfaces, I would not expect FreeSurfer to do a great deal better.