The liar conversation was most likely referencing their previous exchanges; Conan lied about not knowing Rikumichi Kusuda and Bourbon lied about not being an enemy of the bad guys. We have no indication of Bourbon being on to Conan's identity, except that he calls him "a terrifying man" once, but most likely he was just referring to Conan's involvement in the whole Akai business and that Conan is way smarter than he lets on. But beyond that, I'm not sure that Gosho will want to go down that route, so I don't think Bourbon will discover his identity (besides, Vermouth would probably kill him if he found out, besides for his other knowledge).