You can compute them with the FinInG package in the computer algebra system GAP. FinInG has available all the classical generalised quadrangles, elation generalised quadrangles (with a $q$-clan, flock, or Kantor family as input), and it shouldn't be difficult to compute the known Payne-derived quadrangles from the symplectic generalised quadrangles. Here is an example:
 LoadPackage("fining"); gq := SymplecticSpace(3, 5); matrix := IncidenceMatrixOfGeneralisedPolygon( gq );