When State Sen. Edward T. Conroy's pals in the Prince George's County delegation want to get a rise out of him, they send him mail addressed to "Senator Leo E. Green."

Sometimes they tease him about the time his German Shepherd bit one of Green's sons, accusing Conroy jokingly of training the animal to "sic any Green" on command.

Conroy is the senator from Bowie and Green, now a member of the House of Delegates, covets his job. That simple fact is the basis for what has become Annapolis' most celebrated rivalry, a political sideshow that is the object of constant joking and pranksterism as well as serious political speculation.

The two Democrats from Bowie rarely clash in public. instead, they quietly backbite at every turn, each always trying to upstage the other, both within their home district and in the General Assembly. "I never saw anyone go at it like those two guys," said Del. Paul E. Weisengoff (D-Baltimore). "Everything Green does, Conroy wants to know about. There's always that tension."

Bowie Mayor Audrey E. Scott said she had "the toughest job of moderating" when she hosted the city's annual meeting in December with the legislation. "I was sitting between Ed on my right and Leo on my left . . . Leo and Ed outdid each other. Leo would say he'd introduce a bill and Ed would say he was working on that very idea. Then Leo would say he suggested it first."

Meyer M. Emanuel Jr., a former Prince George's County senator who retired this year, said he often found himself acting as a referee when Green and Conroy sponsored the same bills in their respective chambers.

"Green would come over and say 'hey, Manny, please see to it that my bill passes.' And Conroy would ask me to hold up the House bill (Green's) because he had a similar bill he wanted passed."

Sen. John J. Garrity (D-Prince George's) called the sniping an "affirmative type of defensive action."

On the surface, Conroy and Green are unlikely protagonists. They both practice law in Bowie, attend the same Catholic Church, belong to the same civic associations, come from Irish families, graduated from parochial schools, send their children to the same schools and vote the same way on most issues.

When it comes to politics, however, their paths often diverge. The differences in political style and affiliation form part of the basis of their rivalry.

Conroy is an organization man, a loyal member of the county's Democratic organization and the party leadership's Breakfast Club. He is known as a team player.

Green, a popular two-term mayor of Bowie, is a political loner, who rejects organization politics. He ran independently of the organization's Blue Ribbon slate in 1974 and gathered the most votes in his district.

He is one of a few county officials who has not publicly endorsed the gubernatorial candidacy of a Prince George's County native son, Senate President Steny H. Hoyer.

It is Green's intractable independence and success at political solitude that worries Conroy, according to county politicians.

"Senator Conroy has always been a good team worker and he relies on the team spirit," said Garrity. "When he sees the team spirit in a state of being violated, he becomes quite vexed."

Del. B. W. Mike Donovan, one member of the county delegation who enjoys teasing Conroy, said the senator is "real sensitive" about Green because the delegate has "no bond of friendship or loyalty. If he thinks he can win, he'll run."

For Green, independence has not come cheaply. When he entered the House in 1975, he was passed over for the assignment of his choice on the House Ways and Means Committee, which handles important fiscal matters. Committee assignments wre delegated on the basis of loyalty to the organization.

Green often gets short shrift at home. He was the only member of the district's legislative delegation who was not included in the official picture of the groundbreaking for the Bowie Health Center last spring.

The rebuffs have embittered and frustrated Green, whetting his appetite for a run against Conroy. Without allegiance to the orgaization, he said, he will never obtain choice committee assignments or influence in the populous House. The smaller Senate makes room for political mavericks, he said.

"If I stay in politics, I don't want to be in the Mickey Mouse," he explained. "I want to be effective. If I'm going to be effective, I'm going to have to place myself in a more effective role. I'm not a part of the thing that you've got to get along to go along."

"I think Leo's a decent guy," said Conroy, and "most delegates want to be senators." He says he has done nothing to block Green's career but would prefer "seeing all of us working together."

He said he realizes that "you aren't going to change certain peoples' personalities," like Green's, and that if he chooses to take him on in the 1978 primary election in September, he is ready for the challenge.

"Nobody has a license on these chairs," Conroy said. "I've had rivals before. I don't get angry because someone wants to run against me."

"It's a political situation for me," agreed Greeh, "nothing personal. My problem is where am I going in politics."

Conroy and Green are unfairly matched in the legislature. As a three-term senator, Conroy enjoys enormous power as chairman of the Senate Constitutional and Public law Committee. Green wields little personal or institutional power as a freshman delegate.

From his front-row desk in the Senate, Conroy, 48, a bulky man with white hair, black horn-rimmed glasses and Cheshire cat grin, plays and important role in making his committee's report on pending legislation.

Green, 45, a compact man with silver-lined black hair, wire-rimmed glasses and nervous laugh, is a distant voice from his seat in the Prince George's County section at the far end of the House chamber.

Ever suspicious of tampering, each combatant watches his bills with an eagle eye as they pass into the other's chamber. Each has asked colleagues to slow down the other's bills until his legislation has passed, according to legislative sources.

Green openly complains that he cannot get his legislation through the Senate because of Conroy's clubby relationship with fellow senators. When Green's bills come up on the Senate floor, a litter often runs through the ornate chamber.

Some senators found it amusing last year when Conroy rose as floor manager of a Green bill aimed at making state buildings available for public meetings and recreation. Even though the measure flew through the House by unanimous vote and won the backing of Conroy's Senate committee, it failed to muster preliminary approval on the Senate floor. Afterwards, several legislators questioned whether Conroy purposely let the bill die by not strongly defending it. Conroy disputed that contention.