A woman from "an Asian country" fled abroad after dumping the infant, public prosecutors said, adding that efforts were under way to extradite her.
A DNA test confirmed that an Asian man was the father, they added.
The prosecutors said officials had also been charged over the invasive exams carried out on women passengers during the initial search for the mother.
There was international outrage after a group of women flying to Sydney - including citizens of Australia, the UK and New Zealand - complained that they were checked for evidence that they had recently given birth.
Qatar's government said the incident began after a newborn baby girl was found in a plastic bag in a rubbish bin at Hamad International Airport's departures lounge on 2 October. The discovery triggered an immediate search for the parents, including on 10 planes in the vicinity.
Several women on board a Sydney-bound Qatar Airways flight said they were ordered to disembark, taken to ambulances on the tarmac, and told to remove their underwear for an examination.
The women said they were given no information by officials and that they did not have an opportunity to provide informed consent.
Australian Prime Minister Scott Morrison described the incident as "appalling" and "unacceptable".
His Qatari counterpart, Sheikh Khalid bin Khalifa Al Thani, said standard procedures were violated and expressed his "sincerest apology for what some female travellers went through".
On Monday, public prosecutors in Qatar announced that they had filed criminal charges against a number of employees working in the airport security department.
The officers had broken the law "by summoning female medical staff to conduct external examinations of some female passengers", and faced up to three years in prison if convicted, a statement said.
The prosecutors said they had also charged the mother of the child - who survived the ordeal - with attempted murder, and that they were "taking the appropriate legal action within the international judicial co-operation to arrest the fugitive".
She faces up to 15 years in prison if she is extradited and convicted.
"Investigations revealed that the infant's mother, who holds the nationality of an Asian country, had a relationship with another person of the nationality of one of the Asian countries," the statement said.
"The father of the infant admitted he had a relationship with the infant's mother, and that she had sent him a message and a photo of the newborn infant immediately after her birth," it added.
"The letter included her saying that she had [abandoned] the infant she had given birth to and fled to her country."
It is understood that the father is still in Qatar. It is not clear whether he faces any charges.
The baby is being taken care of by Qatari authorities.