Yes, you can. But I don't really hear, "This is what's bugging me," by often. Most people say, "What bugs me is when...," or' "That really bugs me!" Or even, "That kind of stuff bugs me."
I guess the closest thing to "This is what's bugging me," that people say often is, "This is bugging me/getting on my nerves." Or, "You/He/She is bugging me." Or "Quit bugging me."
Keep in mind that "getting on my nerves" is more used than "bugging me" nowadays but you can still use bugging.