Through all the five seasons of Game of Thrones we had never seen a White Walker talking.
Does this mean they can't talk human language (e.g. they have their own language) or we will never see them talking (i.e. they can't talk at all)?
They have a language called Skroth.
Skroth is the language spoken by the White Walkers. It sounds like the cracking of ice.
The language was created for the show, but ultimately wasn't used. Instead the actual ice-cracking sounds were used.
The credits for episodes of Season 1 of the TV series include the statement: "Dothraki, Ashai'i, and Skroth languages created by the Language Creation Society and David J. Peterson."
Peterson developed a spoken language for the White Walkers to use in Season 1 (probably in the unaired pilot episode), but it was not ultimately used on-screen, and dropped entirely from Season 2. Instead, actual ice-cracking sound effects were used when the White Walkers are conversing.
In an interview with Entertainment Weekly during Season 2, Executive Producers David Benioff and D.B. Weiss stated that "Skroth" is indeed the official name of the language used by the White Walkers.
In the books it is stated that they have a language, but its name is never given.
The novel A Song of Ice and Fire clearly states that "the Others" ("White Walkers" in the TV series) speak a language with the voice "like the cracking of ice on a winter lake", but the name of the language is not given.