There are only two ways it could have gotten a fever(as far as I know: Beast sickness or a wound, and the latter is unlikely. FB and Clowns often come with diseases. You can tell by their description if they have one: Beasts with
Noxious Secretions, Poisonous bite or sting, Spit, Toxic blood, vapor, or gas and
Deadly dust can carry syndromes ranging from harmless to instant death. Beast sickness can easily destroy a fort and there is no way to cure it.
If I remember correctly you recently fought one with
Deadly dust, so check the dwarves that engaged it and the place where you fought for coatings of dust.
A start would be to search for where the goat could have gotten the disease, since it's movement is probably easier to reconstruct than your dwarve's.
Contaminants are left by
Noxious Secretions, Toxic blood(I think vapor too but I'm not sure) and
Deadly dust. Contaminants can be found by looking for miscoulored tiles or through loo[k]ing.
To combat the spread of disease, make sure all dwarves are wearing shoes, though full clothing is obviously even better, since most cases of beast sickness are contracted by walking through a contaminated tile. Ensuring your dwarves have a place to clean themselves is also helpful, but be careful, because the contaminants will be washed onto the floor there. Items can also be on items, so if you don't find any contaminated tiles or even after cleaning up people become infected, check frequently used items as well(this is usually food. There is a story somewhere on these forums where one particularly tasty egg roast was contaminated and the player only managed to find it after almost the entire population had bitten the dust(see what I did there?
)).
Contaminated tiles are routinely cleaned by your dwarves but it is a very low priority labor so it can take a while if you don't dedicate some dwarves to it. Cleaning dwarves or dwarves touching contaminated items get infected if they aren't wearing gloves or mittens, so these are important as well. You can also wash contaminants away with water or completely destroy them with magma, but this requires a lot of engineering.
As a last ditch effort there is a DFhack command to clean all contaminants from the map.
It's also good to know the symptoms of your particular case, since not all diseases are fatal.