Doctors had to perform an emergency C-section as they feared Ms Stubbs wouldn't survive the night.
Their baby girl, Quinn, spent the first three weeks of her life in an intensive care unit, while her mother fought for her own life in the ICU.
"They said they didn't think she'd make it through the night, and everything was pretty negative but I knew she would make it," Mr Welsh told Nine News.
"Then they said she'd be brain dead, but she's definitely not."
As soon as Quinn was born, Mr Welsh has made an effort to ensure she gets to spend as much quality time with her mum as possible.
Quinn was discharged from hospital on Monday and though Ms Stubbs is slowly regaining consciousness, she still has a long road to recovery ahead of her.
Mr Welsh said his fiancee's memory has been affected by the event and she doesn't remember getting engaged and sometimes forgets she has a baby.
"I bring the baby to the hospital every day and sometimes she whispers to me that she wants Quinn, and sometimes she asks whose baby it is," he said.
A GoFundMe page has been set up for Ms Quinn to help with her ongoing medical treatment.
"Caitlin's time in ICU was the battle of her life. She had to fight numerous infections in her brain and ongoing bleeds in her brain. She needed help to breathe," the page read.
"Thankfully from the amazing care of the medical team at the Royal Brisbane she has been making slow but significant progress.
"The reality for Caitlin and Jonathon and their baby Quinn is a future filled with uncertainty over a very long recovery. She will need extensive rehabilitation plus future surgeries not limited to but including the removal of the AVM from her brain."
Ms Stubbs will also require intensive physiotherapy, speech therapy and a number of other ongoing treatments.
"Baby Quinn and her Dad will also be needing support as he begins the care of a fragile newborn without the help of Caitlin," the page read.