Don't know if you got the memo, but "the hiiiiiiiiillls are aliiiiiiiiive with the sound of muuuuuuuuuusic."
The Salzburg tourist industry knows these mountains were made famous by the film and puts the mountains and the movie front and center on its tourism website.
The mountains are where Maria goes to sing and commune with nature, to the detriment of her duties at the abbey. When the Reverend Mother worries that she could get lost up there after dark, Maria insists that she grew up in those hills, so she could never be lost there. Basically, the hills are a place where she goes to recharge.
The mountains are versatile actors: they also play the opposite role, representing anything but comfort and safety. Mountains are where you're tested, and where you face challenges that might seem insurmountable It's no coincidence that when Maria is really struggling to find her life's path, Reverend Mother launches into "Climb Ev'ry Mountain" to encourage her not to hide from her problems, but face them head on until she figures out her purpose.
Mountains also present very literal—and huge—challenges toward the end of the film, when Maria and the rest of the von Trapps realize that they're going to have to flee over the Alps to escape the Nazis. The last images of the film show the von Trapps slowly but surely making their way through the mountains into Switzerland. It's the end of a story that began with that opening musical sequence with Maria singing in the Austrian mountains.
The mountains are the same, but Maria's life has changed dramatically. Whereas Maria's girly life at the beginning was all about frolicking in the hills, her adult life with Georg is fraught with huge challenges and dangers—that is, with metaphorical mountains. It seems fitting that the movie ends with her climbing the Alps.
She's definitely learned to face life challenges, and the film's mountain motif illustrates that perfectly.