It depends on which Fossil Pokemon you mean.
The only Fossil Pokemon with released Hidden Abilities are Omanyte, Omastar, Kabuto, Kabutops, Aerodactyl, Lileep, Cradily, Anorith, Armaldo, Cranidos, Rampardos, Shieldon, Bastiodon, Tirtouga, and Carracosta, and they're all only available through transfer from BW or B2W2.
We only know Tyrantrum and Aurorus' Hidden Abilities from the Battle Maison, but they are not available. One can only hope for the days when one can spam Rock Head Head Smash on Tyrantrum...