She refused to meet with them.
The department said it will "consider this referral in the usual way, as it does with all UNHCR referrals".
Ms Alqunun's asylum application was fast-tracked, partly because of security concerns, after the young woman's father and brother arrived in Bangkok and asked Thai police to see her.
Ms 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.
The U.N. High Commissioner for Refugees granted her refugee status on Wednesday.
Qunun alleges abuse by her family - who deny the allegations - and rights groups also said she had renounced Islam, risking prosecution in conservative Saudi Arabia.
"Usually it's really very slow", said Mary Anne Kenny, a veteran legal practitioner and expert in Australian migration at Perth's Murdoch University.
Alqunun's father denied physically abusing her or trying to force her into an arranged marriage, among the reasons she gave for her flight, Thailand's Immigration Police chief said after meeting him Wednesday.
Thai immigration officials had initially said she should return to Kuwait, where her family were waiting.
A UNHCR representative said "the process is still ongoing".
Her online campaign was so successful that Saudi official Abdalelah Mohammed A.al-Shuaibi told Thai officials through a translator: "We wish they had confiscated her phone instead of her passport".
On Sunday Ms Qunun told AFP her family was "abusive" and once locked her in a room for six months just for cutting her hair. She said had been confined inside a hotel room at the airport, under the guard of men she said were from the Saudi embassy and Kuwait airlines. Saudi religious law forbids her to travel alone without permission of her male guardian.
The UN's refugee agency has ruled Saudi woman Rahaf Al-Qunun is a refugee..
He described the man as being a governor in Saudi Arabia.
Al-Araibi's case is being considered by Thailand's justice system, she said.
Her Twitter account gained more than 100,000 followers over six days but the teenager reportedly suspended her profile on Friday after receiving "some very nasty, very real death threats". 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". He is obligated to do everything in his power to advocate, both privately and publicly, and to use the vast leverage that football has, with the Bahrain government, his own government, he's a Bahrainian national, and also with the Thai government to release Hakeem. She had said on social media she was afraid of such an encounter. Bahrain is seeking his extradition.
"I think the most important thing is a young woman in these very very hard circumstances is supported appropriately through the United Nations processes and that countries like Australia who are in a position potentially to provide her with support are able to work from that point once those are finalised and determine what the next steps are", Ms Payne said.