Alqunun, from Hail in northwestern Saudi Arabia, said she feared she may have been killed if forced to return to her family.
A report from the Daily Mail suggested the 18-year-old had been granted asylum in Australia, however, that had not been confirmed by the Federal Government.
Her friends said she had suffered abuse at family members' hands.
She was off to get winter clothes, said Mario Calla, executive director of COSTI Immigrant Services, which is helping her settle in temporary housing and applying for a health card.
Canada's acceptance of Alqunun is likely to further upset its relations with the Saudi rulers.
Ms.al-Qunun's case has highlighted the cause of women's rights in Saudi Arabia.
Canada's ambassador had seen her off at the airport, where Alqunun thanked everyone for helping her.
Australia was assessing a request to resettle her, but hours after the young woman tweeted that she had "some good news and some bad news".
Several other countries, including Australia, had been in talks with the U.N.'s refugee agency to accept Alqunun's bid for asylum.
Protagonista de Guardianes de la Galaxia anuncia compromiso con hija de Schwarzenegger
Pero el pasado 14 de diciembre, con motivo del 29 cumplideros de la joven, Chris subió una bonita felicitación dedicada a su nueva novia, Katherine Schwarzenegger .
Robertson said Thai authorities were spreading fake stories about Rahaf, citing one Thai immigration official's remark to CNN in which he said Rahaf was denied entry into Thailand because she didn't have the requisite documents.
On Friday, Trudeau avoided answering a question about what the case would mean for relations with the kingdom, but he said Canada will always unequivocally stand up for human rights and women's rights around the world.
Alqunun live-tweeted every stay at the airport, posting videos and constantly updating her followers.
Qunun grabbed global attention when she sent out pleas for help via social media in Bangkok last Saturday.
Experts agree the media focus on al-Qunun's case has put pressure on officials in Thailand - which usually have little sympathy for asylum seekers - to act quickly.
"Al-Qunun's plight has captured the world's attention over the past few days, providing a glimpse into the precarious situation of millions of refugees worldwide", said UN High Commissioner for Refugees Filippo Grandi.
Freeland said the United Nations refugee agency found she was in unsafe situation in Thailand and that Canada is glad they were able to act quickly to offer her refuge. Multiple supporters, including journalist Sophie McNeill, who has been in contact with Alqunun during her ordeal, said on Twitter that she was fine but had received death threats.
"Once again we are seeing the abusive influence of Saudi authorities overseas as they seek to forcibly return Saudi women fleeing mistreatment and violence by their families", said Page.
Australian Foreign Minister Marise Payne said Thursday she raised her concerns over his continued detention with Thai authorities during a visit to Bangkok this week.
Australia was her chosen destination because it was possible for her to procure a visa online - "Because I wasn't able to go anywhere alone", she said in reference to Saudi Arabia's guardianship laws, which limit women's movements and freedom to travel.