According to the reports, the woman has been in a vegetative state since she was the victim of a near-drowning more than 10 years ago.
Phoenix police say the matter is under investigation and decline further comment.
The 29-year-old, who is a patient at the centre which cares for people with intellectual disabilities, unexpectedly gave birth on December 29.
Following the azfamily.com report, Hacienda Health Care and state officials issued brief statements about the situation, expressing concern and describing reviews being conducted and steps being taken.
"Bill Timmons, longtime Chief Executive Officer of Hacienda HealthCare, has terminated his employment from the company effective immediately", a Tuesday statement from the facility to HuffPost reads.
They include increased staff presence during any patient interaction, more monitoring of patient care areas and additional security measures involving visitors.
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But Cesena says neither she nor the other parents' families have been given any details about what those extra security procedures are.
"Trust has been severed, period", she said.
KPHO also spoke with a former Hacienda manager who said Timmons insisted that a 1988 incident of abuse of a different patient be covered up.
"I want to assure our patients, their loved ones, our community partners, the agencies we do business with, Governor Ducey and the residents of Arizona, we will continue to cooperate with Phoenix Police and the investigating agencies at all levels in every way possible", said Orman. It's also unclear if staff members at the Hacienda de Los Angeles facility were unaware of the pregnancy until the birth.
"Hacienda stands committed to doing everything in our power to bring this police investigation to a quick conclusion", a hospital spokesperson told ABC Phoenix affiliate KNXV.
Hacienda officials have called the situation "a deeply disturbing incident" and said they are cooperating with law enforcement and state agencies.
Gary Orman, executive vice president of Hacienda HealthCare, said the company would "accept nothing less than a full accounting of this absolutely horrifying situation".
The new mom and baby boy are reportedly both in good physical health.
A male patient complained Hacienda HealthCare staff walked in on him while he was trying to shower with assistance in December 2017.