Her posts attracted global attention including rights groups such as Human Rights Watch and the UN's refugee agency who have both urged Australia to allow entry to Ms Alqunun, who says she is fleeing physical and psychological abuse in her home country.
Australia's Home Affairs Department announced late Tuesday it would consider Qunun's application for asylum if she was found to be a genuine refugee, and called on Thai authorities and UNHCR to assess her claim as quickly as possible.
Alqunun's case has again highlighted the cause of women's rights in Saudi Arabia. People who convert to another religion from Islam risk being charged with apostasy - or abandoning their religious beliefs, which is legally punishable by death.
Her father is now in Bangkok and asking to speak with Qunun, Thai authorities said. Describing her case as a "family affair", it said Saudi officials had neither seized her passport - as Alqunun had claimed several times - nor demanded her deportation back home. Public pressure prompted Thai officials to return her passport and let her temporarily stay in Thailand.
"If my family come, they will kill me", she said in a video archived on Twitter.
In a Twitter update on Wednesday, Ms al-Qunun thanked her 107,000 followers for their "support in my hard psychological situation" and said that she had "regained my strength" after a dramatic few days.
Photos released on Monday night by immigration police showed the teenager with Thai and United Nations officials after she left the airport transit hotel room where she had been holed up over the weekend.
Australia said on Wednesday it would consider taking in Alqunun after the UNHCR referred her case to Australia.
"Any application by Al-Qunun for a humanitarian visa will be carefully considered once the UNHCR process has concluded", the official said.
However, Thai immigration chief Surachate Hakpan said the men would have to wait to learn whether the UN's refugee agency would allow them to.
"She is now under the sovereignty of Thailand; no-one and no embassy can force her to go anywhere", he said.
"Thailand is a land of smiles".
"We will not send anyone to die".
However, officials in Australia hinted that her request will be accepted. She soon started posting messages on Twitter from the transit area of Bangkok's Suvarnabhumi airport saying she had "escaped Kuwait" and her life would be in danger if forced to return to Saudi Arabia. Several female Saudi runaways fleeing abuse by their families have been caught trying to seek asylum overseas in recent years and returned home.
In October, the brutal murder of journalist Jamal Khashoggi at a Saudi consulate in Turkey further heightened tensions and put global scrutiny on the country's human rights record.
The UN's High Commissioner for Refugees granted her official refugee status on Wednesday, according to the Guardian and BBC.
He told Australia's ABC that he was concerned about the arrival of Alqunun's father in Thailand.
A woman in Britain had launched an online petition calling on Foreign Secretary Jeremy Hunt to grant Qunun asylum and issue her an emergency travel document.
In some cases, Saudi authorities were involved in forcing women to return to their families and in other cases, local authorities suspected the women of seeking asylum and deported them, the activist said.