This answer is being updated to bring things into line with the BMM. The exact wording (which will also appear in a future TW printing) is:
"Provided it still has all four legs, and has taken no critical hits to any hip actuator, a prone quad ’Mech fires as though it were standing (the +2 modifier for firing when down does not apply, and it may fire leg-mounted weapons)."
So a prone quad can either fire entirely as if it were standing (all weapons, no +2 mod), or entirely as if prone (no leg-mounted weapons, +2 mod, just as a prone biped mech), but it never splits the difference.
Paul is correct in that any mech in general can fire non-leg-mounted weapons if prone--quad or not, damaged or not--but there was some confusion as to what this meant for quads combined with some other answers so I decided to update this one.