Gibson, 25, was carried off on a stretcher during the first half of Ireland's 3-1 home win over Kazakhstan in their Group C match on Tuesday.
Everton said a scan showed he had damaged his anterior cruciate ligament and the length of his absence "will be best determined once the surgery is completed.
"The scans did confirm to Everton's medical team that no additional damage was sustained to the structure of the right knee, ensuring all treatment and rehabilitation will be focused on repairing the damaged ACL," the club said on their website (www.evertonfc.com).
Gibson has featured in one Premier League game this season under new manager Roberto Martinez. Everton, who host Hull City on Saturday, are seventh on 12 points from seven games.
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- anterior cruciate ligament