Probably not, at least if you don't live in them. Many people pronounce the name with a long "a", sort  of like "Appalaychian". Pronounce it that way in this area, and likely someone will correct you, especially if you are in the home of Appalachian State University. They will certainly know that you are from some other area. One of the first things students at Appalachian are told, no matter their origins, is the correct pronunciation, which is as in "latch", like the latch on your door.
    These mountains were named after a nation of American Indians living in the Appalachian foothills of present day Alabama and holding territory into the Apalachicola area of  present day Florida. They were known as the Apalachee, and the mountains, river and bay all hold their name and keep it alive, though they no longer exist as a nation, though they were hopefully absorbed into the great Creek Confederacy, descendants of whom still live in parts of the Deep South.
    The name of our mountains is pronounced exactly in the same manner as the Apalachicola River and Bay in Florida, and many of us get quite irritated when people come from other areas and tell us how to pronounce the name of the mountains that are our home. Please, when you come to visit us, pronounce our mountains name in a way that will lie softly on our ears.
Thank you.
Alex Netherton
Appalachian Naturalist