Senator Payne praised Thailand for referring the case to the United Nations, which determined Ms Alqunun's was a refugee.
Qunun has refused to meet her father and brother who flew to Bangkok this week, Thai immigration chief Surachate Hakparn said.
"What is truly appalling is how the Saudi Arabian government has acted in sending an official to physically seize her passport from her in Bangkok airport worldwide transit", Mr Robertson said.
"When she arrived, she opened a new (Twitter) account and her followers grew to 45,000 in one day", he said in Arabic.
She was later placed in the care of UNHCR workers while her bid for refugee status was considered.
It noted that Hakeem al-Araibi, a refugee and "torture survivor" from Bahrain granted residence in Australia, has been detained by Thailand since November awaiting a hearing on a Bahraini extradition request. In a video clip of the meeting released by Thai immigration police, Alsheaiby is heard telling Thai officials: "From the moment she arrived, she opened a new account and her followers reached nearly 45,000 in a day. and I would have preferred it better if her phone was taken instead of her passport".
"The Department of Home Affairs will consider this referral in the usual way, as it does with all UNHCR referrals", a spokesperson said.
Rahaf Mohammed Mutlaq Alqunun, 18, claims she was abducted and had her passport confiscated by Saudi Arabian diplomatic staff after she arrived at Bangkok's Suvarnabhumi Airport on Sunday.
The Department of Home Affairs confirmed that the UN High Commissioner for Refugees (UNHCR) had referred 18-year-old Rahaf Mohammed Alqunun to Australia for consideration for refugee settlement.
Before the referral, Australia signalled it would seriously consider allowing Ms Alqunun to settle in Australia after urging the UNHCR to process her case quickly.
Hunt said he had spoken to immigration minister David Coleman about Qunun's case late on Tuesday.
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Reports said it was a diplomat from the Saudi Embassy in Bangkok who seized her passport to prevent her travelling to Australia.
"She does not wish to go back and we will not force her".
Qunun's desperate tweets ricocheted across social media with the #SaveRahaf hashtag drawing an outpouring of support but also the bile of some hardliners in her native country.
"I am giving my family 48 hours (to) either stop or I will publish everything that will incriminate them", she threatened on Twitter.
Saudi Arabia has some of the world's toughest restrictions on women, including a guardianship system that allows male family members to make decisions on behalf of female relatives. "It was due to Saudi intervention that she has not made it".
Human Rights Watch Asia division deputy director Phil Robertson questioned a statement by Thailand's immigration chief to the BBC that Ms Al-Qunun did not have a travel visa for the Asian country, saying she was in transit to Australia when she was detained and did not need a visa, which is available on arrival anyway.
"The government will be making no further comment on this matter".
It praised Thailand for its actions in Alqunun's case, but said the country had not treated other asylum-seekers in the same responsible manner.
"[Ms] Rahaf is not a political asylum case", he insisted.
Ultra-conservative Saudi Arabia has come under fire since the murder of journalist Jamal Khashoggi in the country's consulate in Istanbul previous year.
Crown Prince Mohammed bin Salman had styled himself as a reformer, with women recently granted the right to drive, but these cases raise questions over how the regime exercises control.