The call came after a second anonymous grand juror issued a statement saying the panel was not given an opportunity to consider homicide charges in the case against police officers who served a no-knock warrant at Taylor's home the night she was killed.
The unusual statements -- grand juries are intended to be secret, with some exceptions -- followed a ruling from Jefferson County Circuit Court Judge Annie O'Connell and shed further light on a case that has led to protests across the country and exemplified the ways that the "war on drugs" disproportionately hurts Black Americans.
"The current situation raises serious concerns, as multiple grand jurors are now claiming Attorney General Cameron has not been truthful to the public about what occurred in the grand jury process," Beshear said in a statement. "I trust Kentuckians with the truth, and the next step should be to release all information, evidence, grand jury conversations, recorded or not -- everything."
CNN has contacted Cameron's office for comment.
The 26-year-old Taylor was fatally shot by Louisville Metro Police Department officers who broke down her apartment door during a botched drug raid on March 13.
Two anonymous grand jurors make statements about proceedings
The judge ruled this week that the grand jurors may speak publicly about the case presented by Kentucky Attorney General Daniel Cameron.
"The grand jury did not have homicide offenses explained to them," the first anonymous grand juror wrote in a statement issued Tuesday by attorney Kevin Glogower. "The grand jury never heard about those laws. Self-defense or justification was never explained either."
A second grand juror echoed that statement on Thursday. The second panel member said the jury was only allowed to consider three wanton endangerment charges against one officer.
"No opportunity to consider anything else was permitted," the grand juror said, adding that they agreed "wholeheartedly" with the first juror's statement and looked "forward to continuing to help set the record straight."
The first grand juror said "questions were asked about additional charges" and that members were told "there would be none because the prosecutors didn't feel they could make them stick."
"The grand jury didn't agree that certain actions were justified, nor did it decide the indictment should be the only charges in the Breonna Taylor case," the first juror said. "The grand jury was not given the opportunity to deliberate on those charges and deliberated only on what was presented to them. I can't speak for other jurors, but I can help the truth be told."
Cameron says he asked the panel to indict on charges that could be proven
Cameron, in a statement posted on Twitter earlier this week, said he will not appeal the decision allowing grand jurors to speak.
He said he asked the grand jury for an indictment on charges he thought could be proven in court.
"Indictments obtained in the absence of sufficient proof under the law do not stand up and are not fundamentally fair to anyone," he wrote.
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