When the time finally came yesterday for the Prince George's County Council to elect its new chairman, the council chambers grew suddenly quiet, and for a few long, tense moments, it seemed that the carefully worked-out compromise might fall through.

Council members stared solemnly at each other, doodled on notepads or reviewed their notes, each waiting for someone else to nominate William B. Amonett for chairman and David G. Hartlove Jr. for vice chairman. Finally, council member Francis B. Francois said, "Let's get this done," and shortly thereafter it was over, to everyone's relief.

Amonett became chairman, Hartlove vice chairman and a public seal of approval was stamped on the various accommodations council members had made with each other. In the absence of Winfield M. Kelly Jr., former council member and now, former executive, it was the first step in the all-Democratic council's search for new leadership.

More importantly, it was a step the council managed to take without a public eruption of interparty warfare. With several council members eager to fill the gap that Kelly left, it wasn't easy.

"We have come through a primary election that was a little bit different from four years ago, and a general election that was... also different," council member Gerard T. McDonough said after nominating Amonett. "We have to recognize that the situation is different now."

For eight years Kelly was the recognized political leader of the council, and each year a council member whose turn had been designated years in advance was routinely elected to the council chairmanship.

In any case, the position carried little real political power. When Kelly was on the council, he served as "unofficial majority leader," in Francois words. When he moved up to the executive's office, McDonough replaced him in this role, controlling and guiding the council majority on important votes.

With McDonough guiding the council and Kelly in the executive's office, the Democrats in county government had a working coalition. Then, on Nov. 8, Kelly was gone, beaten in the executive's race by Republican Lawrence J. Hogan. "The coalition loosened," Francois said. "There is no leader and it is all fluid now, with no one person in control on the county government level."

Suddenly, the council chairmanship became important. With a Republican as executive, the Democrats needed a new leader, or leaders, on the council. And whoever moved into that position would be in a good position to challenge Hogan in the executive race in 1982, as Kelly had challenged Republican County Executive William Gullett in 1974.

And so the struggle began, not so much for the chairmanship but for the early position of leadership. Francois, who boasts justifiably that "right now, I would be the consensus choice among Democrats for executive (in 1982), if I choose to run," quickly withdrew from the race.

That left Parris N. Glendening, the ambitious and respected upstart to battle for votes with McDonough, who preferring to remain in his unofficial role as majority leader for the time being, was supporting Amonett for the chairmanship.

The votes swung back and forth for weeks as Glendening and McDonough counted and bargained and wheedled and cajoled. Finally, by last week, McDonough had six votes for Amonett.

But the process didn't end. A balance had to be struck. There was a meeting last weekend, and when it was over, Glendening had agreed to concede the chairmanship, but had been promised McDonough's position as chairman of the Fiscal and Planning Committee, and McDonough's support for the council chairmanship, probably next year.