I asked the Chat person on SPG.com this question, and she replied they do not have non-smoking rooms.

Question: Does the Westin Nagoya have smoke-free rooms on any dates?
All I see is this "Smoke-free rooms not available for your dates".

Answer: They do not appear to have non-smoking rooms.

Sure, there are a lot of smokers in Japan, but rates have been decreasing, and now only 37% of males smoke (less than some European countries). You would think Westin could provide at least 50% non-smoking rooms.