Yes. I'd probably shorten the "I am" to "I'm."
You can't be "in A mountain," you can be "ON a mountain" or you can be "in THE mountainS."
"I'm in the mountains in the north of France, doing some climbing."
"I'm on a mountain in the north of France, doing some climbing."
"I'm in the mountains of northern France, doing some climbing."
I can't give any logical explanation for why "French mountain" or "French mountains" aren't right, but they aren't.
You can say "the French Alps" or "the French Pyrenees" but not "the French mountains."