Julio Obama Mefuman, 51, who also had Spanish nationality, "died in Oveng Azem prison" in the eastern city of Mongomo, the MLGE3R movement said late Sunday.
It gave no date for Obama Mefuman's death but accused the country's hardline regime, without elaborating, of "torture."
The group's name stands for the Movement for the Liberation of the Third Equatorial Guinea and is based in Spain, the central African state's former colony.
A spokesman for the Spanish foreign ministry on Monday confirmed the death, but declined to give details.
Andres Esono Ondo, head of Equatorial Guinea's only authorized opposition party, the Convergence for Social Democracy (CPDS), published a tweet in which he condemned "Julio Obama's death in prison."
He called for an international inquiry "to clarify what happened and ensure that all prisoners have the right to family visits."
Esono Ondo took part in presidential elections last November in which 80-year-old President Teodoro Obiang Nguema Mbasogo was credited with 94.9 percent of the vote.
Obiang came to power in a coup in 1979, ruling since then with what rights campaigners say is one of the most hardline regimes in Africa.
Obama Mefuman's death came less than two weeks after Spain's High Court announced an investigation into his suspected kidnapping and torture and that of a second dissident, Feliciano Efa Mangue, who also has Spanish nationality.
A judicial source said that those two, along with two Equatorial Guinea nationals who reside in Spain, were seized in South Sudan in late 2019 and flown to Equatorial Guinea.
Three individuals are being targeted in the Spanish probe, according to the source.
They are one of Obiang's sons, Carmelo Ovono Obiang, his security director Isaac Nguema Endo and Security Minister Nicolas Obama Nchama.
El Pais newspaper said the dissidents were flown to Equatorial Guinea aboard a "presidential plane" and were repeatedly tortured to extract confessions.
In March 2020, Mangue was handed a 90-year term and Obama Mefuman 60 years for allegedly having taken part in an attempted coup against Obiang in 2017.
On January 4, another of President Obiang's sons, Vice President Teodoro Nguema Obiang Mangue, accused Spain of seeking to "humiliate, interfere with and disrespect our sovereignty."
"These terrorists were detained by Sudanese authorities and extradited to Equatorial Guinea" under an "international arrest warrant," he wrote.