Can you buy Diamonds from other players?

Yes, you can buy diamonds on the Currency Exchange (under the Trade tab).

Search For: Diamonds, Search Offers will show you who currently has diamonds listed for sale.

Alternatively you can Sell Currency: Gold and put in how much gold you're willing to sell for how many diamonds and that'll list up your proposed trade/offer and someone can choose to purchase your gold with their diamonds if they're happy with the rate.