A federal grand jury here is investigating allegations that a group of Washington stenographic reporting firms conspired to overcharge government agencies, which pay $10 million a year for such services.

The investigation came to light when one of the firms, Alderson Reporting Co. Inc., pleaded guilty yesterday to a felony antitrust violation in connection with the probe. The firm could be fined up to $1 million by U.S. District judge George L. Hart Jr.

Special Assistant U.S. Attorney Peter H. Goldberg told Hart that the plea was part of a larger investigation into possible bid-rigging and price-fixing violations by Washington reporting firms going back as far as 1959.

Stenographic reporting firms are used by a wide variety of federal agencies to report and transcribe proceedings verbatim. The firms win their contracts with the agencies through a bidding process, and usually are awarded the pacts on an annual basis.

The charges against Alderson, for example, dealt with contracts the firm had for fiscal year 1976 with federal agencies that included the Nuclear Regulatory Commission.

At least five major reporting firms may have been involved in the alleged scheme, the sources said.

According to sources familiar with the investigation, the grand jury is looking into allegations that the firms violated antitrust statutes by determining among themselves what bids they would make to agencies and then agreeing how an individual firm should bid to win the contract.

Investigators also are interested in allegations that some of the firms may have obtained business by paying off some federal officials, one source said yesterday. However, that source made it clear that the case filed yesterday did not involve such allegations.

Yesterday's plea was the result of an agreement between Alderson and the government that the company's officers would cooperate with the continuing federal investigation. The plea was entered by one of the current managers of ARC, Ira Sharp.

In addition to the other elements of the plea-bargaining pact, Alderson agreed to pay $15,000 to settle all civil claims the United States may have against the firm as a result of the investigation.

The anti-trust investigation began in the Justice Department two years ago and was transferred to the local U. S. attorney's office under Goldberg's direction last year.

Stenographic reporting has become an incresingly lucrative field in the Washington bureaucracy over the past several years, as federal agencies with needs for such services have grown.

The most lucrative contracts are in the regulatory fields, such as the one Alderson had with the Nuclear Regulatory Commission.

The transcription of processes such as administrative hearings can generate exceedingly large fees, and the head of at least one Washington reporting firm is said to be earning hundreds of thousands of dollars a year from federal contracts alone.