Eyre said they consulted with experts and co-ordinators to see if there were any available troops internationally they could send the baboons to, but there were none.
Splitting the baboons up and keeping them in solitary confinement for the rest of their lives would be inhumane, as they are social creatures.
Primate team leader Harmony Neale said the troop were stable for a long time, but in the last few months their hierarchy started to fall apart.
The fighting became more frequent and the baboons started suffering worse injuries.
"In the end it lead to a complete disestablishment and non-existence of any hierarchy within that group."
The four baboons were "living individually" in a shared space, concerned only with themselves.
There was increased aggression, anxiety, a reluctance to be around each other, and the baboons were afraid to eat in case they "got into trouble".
Normal behaviour such and social grooming and positive interactions had totally disappeared, and the baboons' quality of life had plunged.
Neale said there was no one thing that could point to what caused the breakdown.
Animal rights group SAFE chief executive Debra Ashton has slammed the decision, saying the "behavioural and habitat needs of the baboons were not being met at the zoo, which goes to show the flaws of keeping animals in captivity".
"Social structures suffer in enclosed environments and could be attributed to fighting and anxiety for animals. When these social systems break down and there is fighting, vulnerable animals are not in a position to be able to escape as they would in the wild," she said.
But Neale said such fighting also happened in the wild, and could still lead to the baboons being killed. However in the wild there was more opportunity for the baboons to move out of their group.
In the wild, baboons live in troops with between 100 and 700 members, meaning one could move out of their immediate group while still staying within the wider family.
The zoo's baboons could not be sent to live in the wild, as they were born in the zoo.
"If we had left them together they would have killed each other one by one," Neale said.
"The outcome would have been the same, they just would have suffered."
By euthanising them, the zoo was able to make sure they had a "good death", instead of spending their last months living in loneliness and fear, and suffering more pain.
"It was an incredibly difficult decision to come to . . . and it's not one that we take lightly."
The decision was not just made by zoo staff - they consulted with experts outside the zoo first.
"We are the ones dealing with this day in day out, we know it's the right outcome, we know we've done everything we can and that this was the right decision.
"It's an incredibly sad time."
The zoo will not be getting any more baboons.
Eyre said the zoo worked hard to determine which species they should bring in. If they believed they could not give an animal a good life, they would not bring in that species.
"It's no longer a case of we can get some baboons, let's have some baboons."
Wellington Zoo also looked at whether a species they brought in could connect to a conservation programme they could support.
"There are many different factors of why we bring animals into a zoo now," he said.
"Fifty years ago they would have had one or two of everything they could possibly get."
Neale said zoos did "a lot of really good work" around conservation and education and were no longer just there so people could come and look at animals.
She said people would be more interested in supporting conservation for an animal if they could see that animal themselves and build a connection to it.
