The 24-year-old, who memorably reached the semi-finals at Wimbledon this summer, is joined by Cameron Norrie and doubles players Henry Patten and Lloyd Glasspool.
Fery featured in Brtain's Davis Cup play-off against Poland in September 2025, beating Olaf Pieczkowski in straight sets to help seal progress to the 2026 Qualifiers.
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There remains a fifth spot up for grabs, which could go to Jack Draper if the former world No 4 can get back to fitness after an injury-ravaged year.
Captain Leon Smith said: "We know Ecuador will be right up for the challenge, so to be able to call on our top two singles players and two of the world's best doubles players - all of whom have had good results in recent weeks - gives us a great chance of securing the win."
The tie takes place at London's Copper Box on September 19-20, with the winners going through to the Final 8 in November.
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