Brinkley, 58, of Sag Harbor, accused Cook, 53, of Southampton, of falling behind on child support payments. Cook said he didn't pay while he had the couple's two children full time for five months while she appeared on Broadway, but has since paid in full.
Justice Mark Cohen gave both sides the afternoon to reach an agreement, telling them, "This case is one that can and should be settled."
But the parties could not settle it. At one point in the hallway, Brinkley responded to a remark made by Cook's new wife, Suzanne, by putting her hand on her shoulder and telling her, "When you find out he's been cheating on you, I'll be here for you."
"You need to get a new line," Suzanne Cook told Brinkley.
Later, Peter Cook said Brinkley should be grateful he's stepped in as a full-time parent while she "plays."
"Instead of suing me, she should be thanking me," he said.
Brinkley declined to speak to reporter and at one point falsely accused a New York Post reporter of taping the court proceedings.
Her attorney, Thomas Campagna, said Cook has offered a "haze of excuses" for not paying the required child support.
They return to court Tuesday.