President-elect Jimmy Carter plans to name a woman to one of the seven senior staff positions at the White House and the probable choice is Margaret (Midge) Costanza, the vice mayor of Rochester, N.Y., it was learned today.

Costanza, 44, was co-chairman of Carter's New York campaign and is a leader in both the women's movement and in several Italian American societies.

News of her probable appointment emerged as Carter spent a quiet New Year's Eve with his family here.

The President-elect was outdoors all of a long, chilly afternoon quail hunting on land he owns outside his hometown. He emerged from the woods at dusk to tell reporters that his bag of quail and a wood cock was "enough for supper."

He planned to spend the evening at home with members of his family, with a strong possibility he would also visit his mother, Lillian Carter, who is hospitalized in nearby Americus for treatment of an arthritic condition in her leg.

Carter's only scheduled appointment of the day, a meeting with his campaign manager and top talent-scout, Hamilton Jordan, was post poned to New Year's day.

Jordan is coming here, according to Press Secretary Jody Powell, to discuss White House staff arrangements with Carter and to brief the President-elect on choices of Cabinet-members-designate for some of the under secretaries and assistant secretaries in their departments.

In addition to Jordan, Powell and Costanza, the others on the senior White House staff are expected to included Atlanta attorney Robert Lipshutz, as counsel to the President; Frank Moore, a longtime Carter aide, as congressional liaison chief; Stuart Eizenstat, campaign issues chief, as domestic policy coordinator, and Jack H. Watson Jr., the transition planning chief, as coordinator of Cabinet activities.

Costanza's probable role, these sources said, will include relationships with outside groups and coordination of the effort which Powell announced Thursday to develop a variety of methods for Carter to meet regularly with "ordinary citizens."

The only other scheduled event for Carter on New Year's Day is an interview with the editor and publisher of the Americus Times-Recorder, a tradition Carter has maintained for several years. Powell said he expected Carter to join the television audience for the football games, especially the Sugar Bowl in New Orleans, where the University of Georgia meets the top-ranked university of Pittsburgh team.

Carter, in a New Year's statement, expressed the hope "that 1977 will be a time of peace and prosperity for all the world."

More informally, he told reporters yesterday that his resolutions for the new year were to "do a good job as president and get along better with the press."