A victim is taking legal action to force an independent inquiry with power to compel witnesses and the security services to hand over documents.
Gary Hoy was abused by two men who were subsequently convicted.
There have been allegations that a paedophile ring at Kincora was linked to the British intelligence services.
The government has so far refused calls for the abuse scandal at the Belfast home to be included within the scope of the inquiry established by Home Secretary Theresa May and headed by New Zealand judge Lowell Goddard.
The government has said that as child protection is a devolved matter, the right place for the Kincora allegations to be examined is Northern Ireland's Historical Institutional Abuse Inquiry, which has been sitting in Banbridge.
The victims of Kincora claim the Northern Ireland inquiry lacks the powers to compel evidence or witnesses from government agencies.
Amnesty International, which is supporting the victims' legal challenge, says two military intelligence officers have alleged that the security services blocked police investigations into the child abuse at Kincora in the 1970s.
The judicial review is expected to last several days.
Three senior care staff at Kincora were jailed in 1981 for abusing 11 boys.
At least 29 boys were abused at the home between the late 1950s and the early 1980s.
One of the men who was later convicted, William McGrath, is believed to have been an MI5 agent.

The 36-year-old was taken to a London hospital in a critical condition at about 03:45 after he became unwell at an address in Carshalton Road, Mitcham.
He died in hospital shortly after 07:00 GMT and his next of kin have been informed.
A police spokesperson said they were treating the death as "unexplained" and enquiries at the home were ongoing.
It is not believed that any other person was involved.
The spokesperson added that there was "no evidence to suggest that the incident is linked to extremism or terrorism".
Surrounding roads were closed as emergency services including fire engines carrying specialist chemical detection equipment attended the scene.
However, these have since reopened.

William Dunlop, of Billingham, Teesside, was jailed in 2006 over the 1989 killing of Julie Hogg.
It is now alleged he raped one girl and forced himself on to another with the offences said to have taken place in Billingham and at Mr Dunlop's home from the mid-1980s to the early-1990s.
He denies three counts of rape and two charges of indecent assault.
On the opening day of the trial, a jury at Teesside Crown Court heard from one of the complainants who said she was raped on several occasions.
She said she had been left disgusted by what had happened to her and had been unable to tell her parents.
The second alleged victim described having to fight off Mr Dunlop as he tried to kiss her.
Both girls were under the age of 16 at the time.
Mr Dunlop's legal team told the jury the claims had been fabricated and said there were inconsistencies in the timings and dates given.
The trial continues.
A judicial review has begun over the decision to exclude the former Kincora boys' home in east Belfast from a child abuse inquiry being held at Westminster.
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A man has died after he "came into contact with chemicals" at a house in south London, police said.
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A convicted murderer has gone on trial accused of historical sex offences against two teenage girls.