Not sure which aspect that you wanted to check. Both require insertion of "a". “It is a wrong idea/notion that everyone should try to smile all the time. Not everything is a reason for happiness” or “The idea/notion that everyone should try to smile all the time is wrong. Not everything is a reason for happiness” Personally, I'd go with "incorrect" over "wrong". "idea" and "notion" are both ok. The sentence itself is not necessarily sensible. Not having a reason is not a reason to not at least try to smile.

I would say that the second sentence sounds more natural, though I would tweak it a bit by writing "Not everything is a* reason to be happy" instead.