The meeting, which is expected to focus on implementing commitments, especially Pakistan's promises, regarding fighting terrorism in its tribal regions, is taking place without the participation of the Taliban.
The first quadrilateral meet was held between Afghanistan, Pakistan, China and U.S. on the sidelines of the Heart of Asia Summit in 2015, and following that, four other meetings were held in Kabul and Islamabad.
The drone strike was reported days after the Pakistan Army said that it had recovered "safe and sound" a family of foreign hostages from the custody of a terrorist outfit after it received and acted on intelligence shared by the United States.
From Afghanistan, in addition to the Deputy Foreign Minister, a representative from the High Peace Council (HPC) will also attend at the meeting.
In Washington, the talks are being viewed as significant not because they could lead to an immediate truce with the Taliban, but rather because they provide an opportunity to Islamabad and Washington to iron out their differences over a political solution to the Afghan conflict.
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No official statement was issued after the meeting, the sixth such session of the QCG process, which was stalled after Afghan Taliban leader Mullah Akhtar Mansour was killed in a drone strike in May past year. "The meeting is aimed to review Pakistan's commitments on talks (peace) that had been made at previous meetings", said Mustaghni.
Amin Waqad - a close aide to Afghan President Ashraf Ghani, during last week hinted at joining the talks by the Afghan government and Taliban team in Muscat.
"There were some mud-built houses which were being used by the mujahideen (Afghan Taliban fighters)", said a member of the Afghan Taliban, asking not to be identified. Pakistan, however, has been accused of failing to fulfill its promises.
He said he was in Afghanistan to help villagers "who live deep inside Taliban-controlled Afghanistan where no NGO, no aid worker and no government has ever successfully been able to bring the necessary help".
Four-nation talks on peace and stability in Afghanistan started on October 16 in Muscat, Oman.