Is there a symbol to denote, say, $$ABCD$$ is a cyclic quadrilateral? (I very much doubt it.)
$$\square$$ denotes a square. $$\blacksquare$$ is used to end proofs, mostly. It's probably not standard but you could use sonething like: $$\bigcirc\hspace{-4.4mm}\raise{-0.5mm}\square$$ \bigcirc\hspace{-4.4mm}\raise{-0.5mm}\square to denote them, if many of them are talked about. Just present the symbol meaning first. Just put in as a definition of a newcommand in your pdf/book/article etc., so you can refer to the symbol easier. This is more of a LaTeX stackexchange question, as far as creating it.