Iron doesn't come from the mountains
Iron ore was most commonly found in ancient times in 'bog iron' which is deposited in current and former swamplands. Mountainous terrain, especially if it is dry, doesn't have to have many, or any, bogs.
The biggest iron mines in the world don't really correspond to mountains at all. Of Kryvvy Rih and the Kursk Magnetic Anomoly in Russia/Ukraine, Mesabi Range in Minnesota, the Carahas mines in Para, Brazil, and Pilbara, Australia; none of these are in mountainous areas.
There is no reason that your mountainous region has a lot of iron available. If the world is in an iron based timeframe (say, an Iron Age), then your land will be relatively metal poor. Even if it has the materials for making bronze, if there is little bronzecraft in the rest of the world, your land might not have the skills needed to make weapons or other everyday items from bronze.
Copper deposits are generally found in the mountains; however, bronze needs both copper and tin. Tin is pretty rare, and can't just be found everywhere. There are plenty of areas of Earth with no nearby tin, mountainous or otherwise.
If you want an additional reason for your people to not be using bronze, then lack of tin is it. Copper by itself is barely better than wood and stone, given how malleable it is. Your people will probably make do with wood and stone where they can, and import iron where you can't do without it; like swords, blacksmith's tools, carpenter's tools, and ploughshares.