Robles will spend the rest of the season at the DW Stadium on loan from Atletico Madrid, putting pen to paper on Friday morning in time to be available for Saturday's home clash with Sunderland.
The 6ft 5ins keeper impressed last season on loan with Rayo Vallecano and Martinez liked the look of him in a recent two-week trial.
He said: "Joel is a great talent and is in an exciting moment of his personal career.
"His experiences at Atletico Madrid and Rayo Vallecano will help him to bring real competition to the goalkeeping department at the club and we hope this can be a productive period for both him and us. We are delighted to have him on board."
Robles added: "I am so happy to be able to start my career in England.
"The Premier League is a dream for any player in the world and when you see other players from Spain and what they have done in England, it gives us hope that we can be successful here.
"Everyone has welcomed me to the club and now I just have to do my best and compete for my place."