Quick, who has adopted the name Sture Bergwall, has asked the Svea Court of Appeal (Svea Hovrätt) to grant him a new trial in the case of 11-year-old Johan Asplund from Sundsvall who disappeared in 2001.
Asplund's body has never been found.
Quick is serving a life term in a psychiatric institution after being convicted of eight murders committed between 1976 and 1988.
During therapy he admitted to all eight murders along with more than 20 others committed in Sweden, Norway and Finland, often describing how he butchered his victims and in at least one case ate the body parts.
In December 2008, however, he suddenly withdrew all his confessions, saying he had been craving attention at the time and had been heavily medicated by doctors.
Questions over Quick's involvement in the murders have been raised over the years. His convictions were entirely based on his confessions; there were no witnesses or forensic evidence.
The application is Quick's fourth petition for a retrial, with the Svea Court of Appeal scheduled to take a decision in the autumn on the cases of two Norwegian women Trine Jensen and Gry Storvik, who were murdered in 1981 and 1985 respectively.
He was furthermore granted a new trial for the 1988 murder of Israeli tourist Yenon Levi and was acquitted in September last year.
A new trial was also granted for the 1988 murder of a nine-year-old Norwegian girl, Therese Johannesen, but the prosecutor decided to drop the case in June 2011 due to lack of evidence, effectively acquitting Quick.