Ban said the Middle East peace sponsors were looking for "meaningful signs of progress in the coming few months" and reiterated their support for an international peace conference to be held in Moscow this year.
"We are trying very hard to seize on the very favourable political atmosphere" following Barack Obama's, the US president, speech in Cairo earlier this month.
In the speech, Obama pledged to pursue a broad-based, comprehensive peace agreement in the Middle East.
Half a million Jews live in settlement blocs and smaller outposts built in the West Bank and occupied East Jerusalem, territories Israel captured in the 1967 Middle East war.
The World Court based in The Hague has ruled all the settlements illegal and Barack Obama, the US president, has pressed Israel to halt all settlement activities as part of a bid to revive peace talks under which the Palestinians would gain independence alongside Israel.
But Israel says settlements must be allowed to proceed with "natural growth".
Binyamin Netanyahu, the Israeli prime minister, has said that Israel had no intention of building new settlements or of expropriating additional land for existing settlements.
However, he has refused to declare a settlement freeze, which could spark a backlash within his coalition government, many of whose members see the West Bank as a Jewish biblical birthright.
Need for unity
Commenting on the Quartet's appeal, Alvaro de Soto, a former UN Middle East envoy now with the Geneva Centre for Security Policy, said the "Palestinians are divided, they need to be reunified. Unreasonable conditions, such as those that were set by the Quartet about three and a half years ago, should not be insisted on".
"Obama has made it clear that if fists are unclenched, the US would stretch out its hand and be prepared to engage," he told Al Jazeera on Friday.
"It seems to me Hamas is perfectly capable of unclenching its fist and, indeed, making sure others do so as well. You haven't seen any rockets, or mortars, let alone suicide bombers, coming out of Gaza recently.
Alluding to the continued Israeli occupation of the Palestinian territories, de Soto said: "You can't recognise a country while you are being occupied by it. It's just not a request that's operative.
"Without [Hamas], you can't make positive progress and I hope the Quartet will have reflected on this".
Netanyahu gave an important policy speech recently in which he gave some ground over the issue of a Palestinian state following US pressure.
But he laid down several conditions, including that any such prospective state would have no military and that it recognised Israel as a Jewish state.
Responding to that speech, Khaled Meshaal, the leader of the Palestinian group Hamas's political bureau, reiterated on Thursday his refusal to recognise Israel as a Jewish state.
At the same time, he endorsed the idea of a two-state solution, accepting the creation of a Palestinian state within 1967 borders, with East Jerusalem as its capital.
The comments came in an address to supporters in Damascus, the Syrian capital.