Technically, you could call something a "mountain river" but I don't think you're going to find that usage very common because it is so vague and rarely do people characterize rivers based upon their surrounding topography. Like saying, "the plains river Platte..." would sound odd. Better to say "the Belaya river, located in the ... mountains."

Two questions to native-speakers:
1) Doesn't a quote from Burns sound a little odd in this context?
2) I understand "loud-pouring floods" in "My Heart's In The Highlands" as "loud-flowing streams". In other words, "flood" here is a synonym of "river", so "Its (the river's) loud-pouring flood" sounds redudundant to me. No?

1. I would have to see the entire text, but if the Belaya can be characterized as a loud-pouring flood with ice cold water, I don't see any problem. The question is more of style, does the piece lend itself to a poetic reference or not? If you're writing an encyclopedia entry it wouldn't but if you're writing a travel piece it could.

2. I think "loud-pouring flood" here is more to convey a strong image of water forcefully coursing over land.