Prime Minister Shehbaz Sharif of Pakistan and Israel's interim prime minister Yair Lapid expressed their best wishes in phone calls to Sheikh Mohamed.
The two leaders also praised the continuing progress and prosperity of the UAE during their respective conversations.
Sheikh Mohamed thanked both men and offered his best wishes to their countries and people.
The President congratulated Mr Lapid on assuming the duties of prime minister and underlined his commitment to furthering UAE-Israel ties.
Mr Lapid will continue to lead the country in the lead up to elections expected in November.
During the meeting, they explored ways to enhance relations between the countries within the framework of the Abraham Accords, which normalised relations between the UAE and Israel.
Mr Sharif was elected as Pakistan's prime minister in April.
He met Sheikh Mohamed, then serving as Crown Prince of Abu Dhabi, in the UAE capital the following month.
They discussed ways to grow relations between the two countries, opportunities to expand co-operation in various fields, as well as regional and international issues of mutual concern.
Eid Al Adha, known as the festival of the sacrifice, will begin on Saturday, July 9.
It is preceded by Arafat Day, the most important day of the Hajj, when pilgrims gather on Mount Arafat in Makkah where the Prophet Mohammed gave his last sermon.