Walker is accused of ambushing and killing Draper police Sgt. Derek Johnson in September while high on methamphetamine before turning the gun on his girlfriend, Traci Vaillancourt, and then shooting himself.
Walker's attorney Denise Porter informed the court her client is awaiting surgery and asked that his preliminary hearing, at which prosecutors will lay out the evidence against him in the murder case, be delayed until after he has dealt with his medical issues and defense attorneys have received discovery materials from the prosecution.
No date was finalized Tuesday.
Both sides will reconvene in February to discuss the case's progress.
Walker is charged with first-degree felony aggravated murder for Sgt. Derek Johnson's death on Sept. 1. Prosecutors have said that seeking the death penalty for Walker is a possibility.
Walker also faces charges of attempted homicide for allegedly trying to kill Vaillancourt during the same shooting episode, and three counts of discharge of a firearm and one count of possession of a weapon by a restricted person, all third-degree felonies.
Vaillancourt, 34, also appeared in court Tuesday.
In an effort to keep the two separated, Walker appeared in the morning and Vaillancourt in the afternoon.
Another date was set in her case for January in which she is charged with two counts of second-degree felony obstruction of justice for allegedly lying to investigators about what happened on the day Walker shot her and the Draper sergeant.
Valliancourt was hospitalized for nearly a week after the Sept. 1 shooting.