The Basque club made the move after Saturday's 2-1 home defeat to Getafe left them 15th in La Liga, 12 points above the relegation places.
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Former Getafe player Moya, 33, had spent the last three seasons at Atletico as back-up to first choice Jan Oblak.
Alonso, who played for Sociedad, Real Madrid, Liverpool and Bayern Munich and won the 2010 World Cup with Spain, retired as a player last May.
Cristiano Ronaldo hit a hat trick as Real Madrid warmed up for their Champions League clash with PSG by convincingly beating Real Sociedad 5-2.
Real Sociedad are helping fans forget Inigo Martinez following his move to rivals Athletic Bilbao by letting them exchange shirts with his name on.
Athletic Bilbao have moved swiftly to sign Spain defender Inigo Martinez after Aymeric Laporte departed for Manchester City.
Alba got in a spat with one of the home supporters during the second half of Barcelona's 4-2 win at Anoeta when he made the gesture.
Barcelona fought back from two goals down at Real Sociedad to secure their first La Liga win at Anoeta in 11 years.
Carlos Vela has achieved what very few Mexican players in history have accomplished in Europe, but there is a sense of lament about his exit.
Major League Soccer-bound Carlos Vela bid an emotional farewell to Real Sociedad after scoring in his last game for the club.
Sevilla missed the chance to break into La Liga's top four as they slipped to a 3-1 defeat at Real Sociedad -- Carlos Vela scored in his final game.
Carlos Vela will bid farewell to Real Sociedad on Wednesday ahead of his move to Los Angeles FC, and he admits the occasion will be emotional.
From the bertso derbia off the pitch to events on it, Saturday's showdown means a great deal, not least to two clubs struggling for their best form.
Atletico Madrid cut the gap on Barcelona to six points with a 2-1 comeback win against Real Sociedad.
Girona and Real Sociedad shared the spoils from a 1-1 draw on Friday night.
Adnan Januzaj and Radja Nainggolan are the high-profile call-ups to the Belgium squad ahead of friendlies against Mexico and Japan.
Asier Illarramendi scored his first-ever goal for Real Sociedad at the Anoeta before being dismissed in a 1-1 draw with Espanyol.
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