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Is there somewhere a database of incidence matrices of generalized quadrangles that one can download?

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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:

gq := SymplecticSpace(3, 5);
matrix := IncidenceMatrixOfGeneralisedPolygon( gq );

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Thanks, I'll try this! – Felix Goldberg May 13 '13 at 19:33

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