I'm not familiar with that forklift, but......if the truck has been equipped with an aftermarket "loss of engine oil pressure" shutdown system.......that system might be why the horn blows after the ignition switch has been turned on for a number of seconds with the engine not running (or running but sensing a loss of oil pressure). One aftermarket shutdown system I am familiar with is configured to sound the OEM horn a couple of seconds before the shutdown is activated to kill the engine.
Other than that idea, I have nothing else to offer.