The deployment made possible by legislation fast-tracked by Turkey following a diplomatic rift between Doha and four other Arab states.
Two days after the crisis broke out, the Turkish parliament swiftly ratified a deal to deploy Turkish troops to Qatar that was signed in late 2014, in a sign of showing solid support to Doha.
Elsewhere in his remarks, Gargash said he believed that when Qatar did back down, there would be a need to monitor its activities in the region, something Western powers could undertake.
The exercises had "been planned for some time", it added.
Saudi Arabia, the United Arab Emirates and Bahrain, plus nonmember Egypt, accused Qatar of harboring, funding and championing Islamist terrorists, in part through the Doha-based satellite news channel Al-Jazeera.
Reconciliation between Qatar and its neighbours is becoming increasingly hard, with the peninsular state seemingly unwilling to back down or concede accusations that it supports terrorism.
Simmering tensions between the US partners boiled over this month when Bahrain, Saudi Arabia and the UAE, along with Egypt, severed ties with Qatar over its alleged support for extremism and imposed economic restrictions on it. Saudi, Emirati, and Bahraini families are being forcibly recalled by their governments today, despite being invited to stay by the government of Qatar. The Qatar Airlines also finds itself in trouble as it is no longer allowed to use Saudi, Bahrainian and Egyptian airspace.
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Sheikh Saif bin Ahmed Al Thani, director of Qatar's Government Communications Office, said in a statement that the blockade, now entering its third week, is tearing at the social fabric of the Gulf.
"Along with Turkey, it is the country with the most resolute stance against Isis which has caused grave damage to our region".
Abadi had been due to visit Riyadh last week but postponed his trip to avoid appearing to take sides in a diplomatic dispute that erupted between Qatar and several other Arab states including Saudi Arabia, officials have said.
Qatar shares its only land border with Saudi Arabia, and relies heavily on food imports, much of it from Gulf countries. Earlier this month Turkish President Tayyip Recip Erdogan denounced the isolation of Qatar calling it "against Islamic values".
Foreign Minister Mevlut Cavusoglu has been one of the figures trying to forge a diplomatic solution to the crisis.
The Qatari Foreign Ministry confirmed on Sunday it has received a verbal message from Iranian President Hassan Rouhani, giving no hint of the content.
The chairman of the U.S. Joint Chiefs of Staff said on Monday that military operations against Islamic State from Qatar were continuing while acknowledging some friction.