Yes they can, although the canine version of gout is not identical to the human one. In humans, it’s a buildup of uric acid in the bloodstream called hyperuricemia that, over time, deposits crystals of uric acid in the sinuvial fluid around the joints. This causes a specific kind of arthritis. In dogs, as far as I can tell, the definiition is extended to any mineral-like deposits on the joints, in the sinuvuial fluid, or anywhere nearby. However, either of these diagnoses are hard to make without a serological study being performed.Note that in humans and dogs, this condition heralds potential for kidney disorders.Unless you have cause to specifically suspect canine gout over trauma or a more general arthritis, I would inspect for trauma and treat for pain.In any case, if this condition has persisted for over a few days, you need to get the dog to a vet.