Note: I don't have any experience with mountain goats in particular, but have hunted and tracked wild goats from other species. I am assuming that they are similar.
Like all goats, there will be signs of their presence in their habitat. The trick is to distinguish this from other animals (e.g. deer) that might inhabit the same area. With mountain goats you are most likely to see them in the upper alpine areas, where there won't be too many other species that you could confuse them with. Where you find one goat, you are likely to find others as they generally exist in herds/flocks.
Signs of goat species include:
- Close (i.e. down to the ground) cropping of feed species (shrubs, grasses, herbs) in the area. Most deer species don't do this in my experience.
- Droppings will be smallish localized piles of rounded pellets, sometimes clumped together to form a more solid mass. Deer feces tend to be a larger oval pellet and more frequently clumped.
- Fur - white fur stuck to broken off branches and twigs. It will be longish and I doubt easily confused for other animals. Deer are generally brown, but can have pale yellowish belly fur...
- Hoof prints - on soft ground between rocks (for this species). Goats are cloven hoofed, as are deer etc. The foot prints will be symmetrical and not parallel (for the cloven bits, not a series of prints), and kind of like a curved drop of water (kidneyish?) shape. Google to get an idea, compare with other animals.
- Bleating - goats bleat, but you are most likely to hear this from kids, and only when they are separated from the adults during the birthing season. Deer have a range of sounds from different species.
- Smell. Male goats in particular smell quite strongly and use this to scent mark territory.
- Bedding - they often bed on sheltered sites that are warmed by the morning sun (they aren't stupid).
There is lots of good information at the USDA page.