Where do emeralds appear most often?

First of all Emerald ore does not spawn in every biome. The wiki says that up to 11 blocks of emerald ore spawn per chunk inside any type of mountain biome on y-levels from 0 to 32.

But if you want to get emeralds and not the ore itself, you are better of just breeding some villagers an building some ridiculous farms that you leave running for some time, so can trade for emeralds.

For example a general mob farm will grant you rotten flesh to trade with clerics, or a sugar cane farm can produse masses of paper for trading with librarians. If you build both you have practically unlimited stock of rockets (added bonus).

I should also mention some "exploits".

If you get a few librarians you will have some with a trade [ emerald => bookshelves ] and some with [ books => emeralds ]. You can cycle through those two, destroy the bookshelves to get books and sell those back to the villagers. This results in about two times the amount of emeralds you invested in the bookshelves in the beginning. This whole thing depends on the prices your villagers offer you, which can be drastically reduced by letting them get converted to Zombie villagers and cure them after that.

So you have something like:

• Step 1: get villagers.

• Step 2: make them librarians.

• Step 3: Infect and cure them (quite expensive for the gold you will need, but... )
• Step 4: PROFIT!!!