You mean, to describe such nonsensical situations in general, rather than to actually talk about this kind of theatre? Then, no, I think most people do not recognize the word 不条理(演)劇, and I have never heard or seen "不条理劇みたいな会話をした" or something like that.
The first word that came into my mind to describe such a situation is シュールな, which is a loanword from "surrealism". I don't know the usage of this word in English, but Japanese people use シュール often (at least more often than 不条理な or ナンセンスな) to describe something nonsense yet funny, impressive but difficult to interpret. It's too difficult for me to fully convey the feelings of this word, but don't believe the dictionary definition and try googling シュールな会話 or シュールなマンガ.