"Game design" is an incredibly broad field that likes to include level design, systems design, and story writing (among many others) under its umbrella.
When I think of game designers, though, I think of systems designers: the people who write the specs that get implemented by programmers. And yes, spec-writing is a skill that is used in the greater software industry (these people are sometimes called "systems analysts" or "program managers" or lots of other names) - just look at all of the "design" articles on joelonsoftware! This goes triple for designers who specialize in UI, since the principles of designing a good UI are still largely the same whether you're making a game or a business application. So, yeah, the skills transfer quite nicely.
One caveat: not everyone in the software industry knows what it's like to work in game development. There is still the stigma of "oh, you used to get paid to just sit around and play games all day? We actually do Real Work here" that has to be overcome in some companies, so you may have to convince people that a game developers is actually one of the hardest workers and most skilled labor they are likely to find. This might mean, for example, carefully crafting your resume and cover letter to focus on job responsibilities that sync well with a given posting, and downplaying the "games" aspect.
