Where did this trope originate from and why sneezing? It must be something only in Asian culture because I have never seen this in any US television shows. And yet sneezing when being talked about occurs in both anime and dramas.
This is something that got featured in the Book of songs, this was a poem book out of ancient china (1000BC) and still is occasionally found in Anime/Manga now a days
In certain parts of Eastern Asia, particularly in Japanese culture and Vietnamese culture, a sneeze without an obvious cause was generally perceived as a sign that someone was talking about the sneezer at that very moment - a belief that is still depicted in present-day manga and anime. In China, Vietnam and Japan, for instance, there is a superstition that if talking behind someone's back causes the person being talked about to sneeze; as such, the sneezer can tell if something good is being said (one sneeze), something bad is being said (two sneezes in a row), or if this is a sign that they are about to catch a cold (multiple sneezes). source
Small side note to "It must be something only in Asian culture because I have never seen this in any US television shows". Although not used as often in movies/series nowadays, they still feature this trope once in a while and is mainly used in a comedic sense. A sample would be Scrubs and NCIS where it got featured a few times.