The language of Rivendell was Sindarin. Sindarin was the language of all the Elves of Middle-earth at the time of the War of the Ring, with the possible exception of the Silvan Elves of Mirkwood ... Tolkien gave conflicting accounts about what language they spoke. Here's what Tolkien said about the language of Rivendell in his notes on A Elbereth:
"The language is Sindarin, but of a variety used by the High Elves (of which kind were most of the Elves of Rivendell), marked in high style and verse by the influence of Quenya, which had been originally their normal tongue." (The Road Goes Ever On, p. 72)
We are told in the appendices of LotRthat Quenya was no longer the daily tongue of any of the Elves of Middle-earth, and was reserved for matters of lore and ceremony by the High Elves.