The France international underwent a lengthy operation on Tuesday which finished at about midnight, just after Barca's 4-0 La Liga victory at home to Getafe during which the whole stadium chanted Abidal's name and applauded the full-back.
"Hi Everybody, I'm Hayet Abidal, Eric's wife. I'm writing you, today, on his facebook page, on his request," she wrote.
"Eric and his cousin Gerard [the donor] have indeed been operated yesterday in Barcelona and for the moment, everything is ok!
"He really wants to thank all of you for your messages, your support and all the strength you give him! Thanks a lot."
Earlier on Wednesday, Barcelona president Sandro Rosell had given the first news on how the 32-year-old was doing as he recovered in hospital.
"Today could be a great one all round, because we know that Abidal and his cousin are recovering well after the operation," Rosell said.
"I can't say anything more than this because the family have specifically asked us not to say any more."
Abidal's Barca team-mate, Spain midfielder Sergio Busquets, wished him all the best when he spoke at a news conference after training on Wednesday.
"The operation is over and the next few hours are very important," he said. "In two or three days we'll see how it is all going.
"From here I went to send him all the very best, as everyone else has been doing."
Abidal had surgery to remove a tumour from his liver last March before making a triumphant and emotional return to lift the Champions League trophy at Wembley just two months later.
Last month, the club said Abidal would need a transplant but did not say how long he would be out of action.
