It would be completely fine. "Bless you" does originally come from a cultural background. However, in the modern day it's just a phrase, nothing more. You'll be fine :)

Hello Shouichi, As far as I know the origin of this expression goes back to the time of the Plague, a cough or sneeze was the first sign that somebody had this illness. It is a shortened form of 'God bless you' as someone who had the first signs was very likely to die. Nowadays it is in general use regardless of religion. Hope this helps Bob

Whether or not you should say "(God) bless you" varies greatly from person to person. Even if some people are of the same religious or cultural background, they may not be comfortable with you saying it. Similarly, some people of different backgrounds may be totally fine with you saying it. Personally, I avoid saying "Bless you" to anyone unless I know that it is something that they are comfortable hearing (friends and family), as I don't wish to accidentally offend someone. Ultimately, however, it is your choice. As an alternative, you may be able to say "Gesundheit!", which comes from German and means something along the lines of "health"/"be healthy" ( I don't speak German, so I'm not sure of the precise meaning).