It's not a big deal.
Your garage will just need to be built into the hill and correctly designed and constructed, just like the other ones nearby.
We still need some information, to be able to tell where and what you can build. What is your zoning and exact location?
Which direction do you want to access the garage from? The uphill side or the lower side?
Are you allowed to create a curb cut on the downhill side?
If so, you could just build a garage, nestled into the hill, and access it from below. You would need steps in the hill, to get up to the house. You could either build an extra height structure to accommodate a lift, or add a bonus room over the garage space, that would have access from the top of the hill.
If you need to access it with your vehicle, from the existing house side, then the structure needs to be constructed in one of two ways.
Either create a foundation and fill it and pour a slab on the fill to create a pad on which to build the framework, or dig into the hill and create a lower leven space out of engineered masonry or concrete, topped with precast concrete slabs, and then frame your garage on that.
This will be the most expensive option.
Do you understand?
