Can thawing meat too quickly affect its quality?
I'd be more concerned about the meat's safety than taste/texture. "Warm" thawing creates the perfect environment for bacteria growth.

That having been said, my experience mainly involves the use of a microwave to thaw meat. When microwaving meat to thaw, try to avoid the meat having corners or protrusions. This adversely affects taste/texture because they usually start to cook before the rest of the meat thaws. 

Freeze ground meat, in a circular shape with a hole in the middle for quickest microwave thawing time. When thawing meat 'parts' place them in a circle, thickest side outward.

So long as no noticeable cooking has started, the taste/texture should be fine. Cook immediately.