The answer is a "yes", but with some qualifiers. Full suspension is heavy. It is going to be a lot more work to drag it up the hills. If you don't have full suspension you will do better.

Go for the most road friendly (smooth, small diameter) tires your rims will take - but train a lot of the summer on the nobbies to build muscle.

You do have a bonus - the low gearing and triple chainrings most mountain bikes have are a real help on hills.

Don't expect to be fast or the first into camp. You will spend a large part of each day on the road. But, as you pedal along, you will also have time to gawk at the scenery and be a tourist. This can make it a real enjoyable vacation. You will have lots of company!