The two men had known each other for years around Capitol Hill. Both were hawkish supporters of Israel and active in the Republican Party. But now they were on opposite sides of a remote dispute between wealthy Persian Gulf monarchs: One was a lobbyist for Qatar, and the other had sided strongly with rivals who sought to blockade the tiny emirate.

“Be very careful,” the lobbyist, Nicolas D. Muzin, warned on March 5, in a conversation quoted Thursday in a legal complaint accusing Qatar of hacking into the computers of its American opponents.

Anyone Mr. Muzin pointed to was “in danger,” he told his old acquaintance, Joel Mowbray, who appears to have recorded their conversation about the Qataris. “Honestly, I know they are after you and after Broidy,” the lobbyist continued, referring to Elliot Broidy, a top Republican donor, critic of Qatar and Mr. Mowbray’s associate.

Their conversation occurred just days after The New York Times and other news organizations began publishing articles based in part on leaked emails from Mr. Broidy’s account.