"It's a big blow for us because he started this game good," Bilic told the BBC of a player who only returned to first-team action in September after a seven-month spell on the sidelines following knee surgery.
"I don't understand it because we're resting the players. We're not training a lot. We can only hope it won't last that long."
In Carroll's absence, West Ham were reliant on the talents of midfielder Dimitri Payet, who made his first start since early November after he picked up an ankle injury.
The 28-year-old Frenchman came up with a goal and an assist to help his team move up to fifth in the league table.
"I'm very happy because I don't play for two months," he said, quoted by the club's website (www.whufc.com).
"I'm very happy I can play, and (get a) good goal, good assist. It's my job. I'm very happy."
(Reporting by Simon Jennings in Bengaluru; Editing by John O'Brien)
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