Minibus driver Ahmet Suphi Altindoken was jailed for life without parole in December 2015 for murdering and attempting to rape Ozgecan Aslan, in a crime that sparked nationwide protests.
Altindoken's father, Necmettin, 51, who was one of two men jailed as accomplices to the murder, was also shot and wounded in the attack at the high security prison in the southern Adana region, the Dogan news agency reported.
Ahmet Suphi Altindoken, 27, was hospitalised after the attack but doctors were not able to save him. The father's life was however not in danger, Dogan added.
The circumstances of the attack and who carried it out are being probed and prosecutors from the Adana region have launched an inquiry.
Ozgecan Aslan
Ahmet Suphi Altindoken was shot close to his heart and a 6.35 millimetre pistol was used, the report added.
During the trial it was revealed that Aslan had been travelling on Altindoken's minibus, and when all the other passengers had got off he drove to a wood and tried to rape her.
The young woman fought back using pepper spray but Altindoken then bludgeoned and stabbed her to death.
Altindoken's father and friend were found guilty of helping him burn and dispose of the body. The remains were found by police and the three were arrested.
The trial was hailed as hugely symbolic in a country where an often-silent wave of violence against women sees hundreds killed at the hands of men, often their husbands, each year.
Killers have frequently been able to secure a reduced sentence by arguing that a woman provoked them, or that their dignity was impugned.
Activists say remarks by government officials about women and how they should be treated leave them exposed to violence.
