President Carter found himself engulfed in help from an unexpected quarter yesterday on his government reorganization bill. Seven House Republicans, led by Minority Leader John J. Rhodes (R-Ariz.), rushed in to sponsor the Carter bill House Government Operations Committee Chairman Jack Brooks (D-Tex.) has declined to introduce for the President.

"I just happen to agree with him," Rhodes said of the unusual development. "I think the government needs to be reorganized, and I think the President ought to 'have that authority."

More directly, Rep. John N. Erlenborn (R-IIl), one of the six Government Operations Committee Republicans who joined Rhodes in the move, said "It was a happy confluence of good legislation and good politics."

The Republican move came 24 hours after Carter had met separately in coffee sessions at the White House with Brooks and Rep. Frank Horton (R-N.Y.), ranking Republican on Government Operations.

The meeting with Brooks produced no timetable on hearings on the bill and no softening of Brook's adamant opposition to the key future of the Carter legislatin: one that would allow a reorganization plan to take effect automatically unless vetoed by one house of Congress within 60 days.

Brooks had introduced his own bill providing that a reorganization (See REORGANIE, A6, Col. 1> plan can become effective only when affirmatively approved by both houses of Congress.

But Carter had better luck with Horton, who said yesterday, "I assured him of my cooperation and said we ought to have hearings as soon as possible."

When Horton told GPO colleagues of his conversation and commitment, several suggested it would be a good opportunity for the Republicans to dramatize their "cooperation" with Carter and , incidentally, point up the Democrat's discord on the Democrat's discord on the issue.

Meanwhile, a group of House Democrats, led by Rep. Elliott H. Levitas of Atlanta, was beginning to fret that four days after Carter had sent Congress his reorganization authority bill - the first step in fulfilling one of his major campaign pledges - the bill still lacked a House sponsor.

After several calls to the White House congressional liaison office, which Levitas said "had not really focused on the problem," a meeting with Carter was arranged. Levitas and House speaker Thomas P. (Tip) O'Neill Jr. (D-Mass.) led a dozen members of Government Operations to the Oval Office.

The result was that 17 Democrats from the committee were recruited "rather quickly," Levitas said, as cosponsors of the Carter bill, thus avoiding the situation of having only Republicans backing the President's legislation.

Together, the two bills are sponsored by 24 of the 43 members of Brook's committee. But the chairman remained unimpressed.

"With coffee at $3.50 a pound," he said yesterday, "it's amazing what results you can get with a cup of coffee."