Russian Foreign Minister Sergei Lavrov, after a Wednesday meeting in New York with the other Quartet representatives - UN Secretary-General Kofi Annan, and the foreign ministers of the European Union and the United States - called on Hamas to cooperate.
"We call on representatives of Hamas, with whom Abbas is establishing contacts, to take as constructive a position as possible, and listen to the Quartet's opinion... in our individual work with Hamas, we are actively sending these signals," Lavrov said.
The Russian foreign minister said this entailed the need to respect all agreements that the sides had reached, and the authority of Abbas in talks with Israel. "We hope his efforts will be rewarded with success," the minister said.
The Palestinian leader's Fatah party was defeated by Islamist Hamas in parliamentary elections in January. Abbas is now attempting to form a coalition of Fatah and Hamas.
Lavrov also said the UN resolution on Lebanon was being implemented and practical progress was being made each day.
The resolution, which led to a truce between Israel and Lebanon-based militant group Hizbollah after a month of fighting which inflicted huge damage on Lebanon, calls for 15,000 UN troops and a similar number of Lebanese servicemen to guard the south of the country.
"We hope that in the very near future, the Israeli army will leave the south of Lebanon. UN peacekeepers are being deployed there at a fast rate, and we also reiterated the need to step up aid to Lebanon in rebuilding its infrastructure," the minister said.
Deputy Foreign Minister Alexander Saltanov, commenting on the Quartet's meeting at the UN headquarters in New York, said Hamas should demonstrate its readiness to cooperate with the conditions and demands set out by the Quartet.
The Quartet had previously put down three conditions that sides in the conflict must fulfill: an end to violence, recognition of Israel's right to exist, and the resumption of the roadmap peace plan.
Abbas met with U.S. President George W Bush on Wednesday and urged him to reactivate the peace plan.