President Reagan is expected today to name Adm. Carlisle A.H. Trost, commander of the Atlantic Fleet, as chief of naval operations, succeeding Adm. James D. Watkins, whose tour ends June 30.

The chief of naval operations, like the heads of other services, serves a four-year term and is one of the Joint Chiefs of Staff. The Senate must confirm the nomination, but no opposition to Trost is expected.

Trost was the choice of Defense Secretary Caspar W. Weinberger and Adm. William J. Crowe Jr., chairman of the Joint Chiefs of Staff, according to Pentagon officials, while Navy Secretary John F. Lehman Jr. was pushing Vice Adm. Frank B. Kelso II.

Sen. John W. Warner (R-Va.), a former secretary of the Navy, said he lobbied heavily for Trost because when the admiral was his aide he found him "one of the most intelligent, fair and independent-minded officers I had ever met."

Lehman, after learning that his choice had lost out to Warner's, phoned yesterday to inform the senator that Reagan had chosen Trost. "That was a gentlemanly gesture," Warner said.

Defense Department officials said that during the selection process Trost impressed Weinberger with his breadth of knowledge and sophistication.

Weinberger met with Kelso in London when the vice admiral was commander of the battle group that crossed the "line of death" in the Gulf of Sidra off Libya in March and bombed the country in a night raid on April 15. Weinberger, according to aides, was impressed with Kelso's understanding of his orders for those operations, but picking Kelso would have meant bypassing four-star admirals to pick a three-star, and Kelso was not considered as well-rounded as Trost.

Trost, 56, was graduated first in the Naval Academy class of 1953 and has been commander of the Atlantic Fleet since last October. Like Watkins, Trost is a submariner.

In what would be another big change at the top of the Pentagon, Donald A. Hicks may resign as undersecretary for research and engineering if Reagan follows Weinberger's advice and appoints Richard P. Godwin, 64, executive vice president of Bechtel Group Inc. and vice chairman of one of its operating companies, Bechtel Inc., to the newly created position of undersecretary for acquisition. Before becoming secretary of defense, Weinberger was general counsel for Bechtel Group Inc.

Because the new undersecretary would be assigned many of Hicks' present duties, including setting research policy, Hicks is said to want the appointment. Hicks has been in his present job for only a year but in that time has assembled a new team in hopes of solving some of the procurement problems that have plagued Pentagon leaders and emboldened Congress to cut the defense budget.

Hicks has picked up some support in the White House to become undersecretary for acquisition, administration officials said, but Weinberger's list has only one name on it: Godwin's.