There are two types of quadrangles in a projective plane, Fano quadrangles, and non-Fano quadrangles. The number of quadrangles in some small projective planes is counted according to type, and an interesting configuration in the Hughes plane is displayed.
Citation
Ralph G. Stanton, William Kocay. Fano Quads in Projective Planes[J], Ars Combinatoria, Volume 057. 217-223. .