Christian and Moslem militiamen today exchanged mortar fire across Beirut's dividing Green Line. No casualties were reported. The two-hour battle in downtown Beirut came amid reports of growing tension between a Christian militia and the Lebanese army. There was no word on what caused the battle, the fourth between civil war antagonists in Beirut in two weeks. The clashes have strained a 14-month-old truce in the war that began in 1975. The fighting coincided with a state of alert proclaimed by the 6,000-strong Lebanese Forces, the Christians' main militia, in east Beirut and the Christian heartland north of the capital. Christian militiamen set up checkpoints and roadblocks all over the Christian sector of Beirut and along the main coastal highway running north from the city, witnesses reported. Fighters in battle fatigues and carrying automatic weapons searched cars and frisked occupants, said the witnesses, who spoke on condition of anonymity. The militia gave no explanation for the alert. But the Christian-owned Ad-Diyar daily said it stemmed from tension between the militia and the Lebanese army, which is controlled by Christian officers. The newspaper said the militia was putting on a muscle-flexing show to demonstrate it controls the Christian enclave, not the army. The commander of the 37,000-man army, Gen. Michel Aoun, has been quoted by several Beirut newspapers recently as saying the military could finish off the militia any time it wanted. Newspapers said his remarks were part of a campaign by the Maronite Catholic commander to become the next president of Lebanon after President Amin Gemayel's six-year term expires in September. Aoun, who has not denied the statements, has not offically proclaimed his candidacy.