Fuses don’t have any moving parts, so it shouldn’t be a fuse that’s making a ticking noise. But fuse boxes also often house relays which do have electro-magnetic contacts and a small armature that moves when the magnets are energised. That’s the probable cause of the ticking noise.
That you’re hearing to ticking noise when you try to start the car, suggests to me that it’s a relay connected with the starting procedure; possibly something to do with powering up the fuel pump or even the starter-motor itself. But hearing it means that it’s working and the problem is somewhere downstream of the relay.
You’ve told me that the car won’t start, but does that mean it cranks and won’t fire, or won’t even crank in the first place? Because those are two very different problems with a thousand potential causes.