Nizar Haraki, the opposition envoy, said this would help thousands of Syrians who fled the country to escape the conflict, which has killed at least 200,000 people since it erupted in 2011.
It was not immediately clear whether the service would extend to Syrians living elsewhere.
About half the country's people have fled their homes since the war broke out, with the United Nations saying 3.8 million of them are refugees, mostly in neighbouring countries.
Haraki said 52 countries had been contacted to ask if the passports would be accepted and that all but one, Canada, had said yes.
Syrians in Qatar have two months from Thursday to apply for new paperwork to renew their passports.
Last year it was estimated that some 40,000 Syrians had Qatari residency, while another 20,000 were on temporary visitors' visas.
Haraki said he did not know how many people would apply, but dozens of Syrians went to the office Thursday.
"This project is to break the siege of injustice and loss imposed by the regime on the Syrian people, and allows every citizen to move freely and to provide a decent living for himself and his family," added Haraki.
Many Syrians have lived in Qatar for decades, but their numbers have increased because of the conflict. Qatar has relaxed its immigration rules to allow Syrians to in the country long term.
Qatar has been one of the main supporters of the rebellion against Syrian President Bashar al-Assad and has allowed demonstrations against him to be held in its capital, Doha.
In February 2013, the emirate handed the Syrian embassy building in Doha to the Syrian National Coalition, the main opposition group, which was formed in Qatar in 2012.
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