Two weeks after giving birth, the 29-year-old shared a photo to her 100,000 Instagram followers of her keeled over in pain during labour.
Alex revealed she was told by her doctors that "at this stage, baby is better out than in" and had to abandon her natural birth plan.
The mother-of-two said she was instantly in "tears and angsty" about the situation.
"But I was meant to go into labour naturally. Induction wasn't part of the 'plan'," she wrote.
"After multiple growth scans, static fundal height and reduced movements, induction became my reality and it was time for us to meet our baby."
Alex shares her 10-year-old son Elijah with her former boss, but Maxwell is her first child with Carson. His middle name Keegan is after Carson's late brother.
Last week Alex revealed that Elijah loves having a sibling, and praised him for quickly adapting.
"He is a such fantastic big brother and helper. I feel so lucky," she said.
She described Maxwell's entrance to the world as "intense and beautiful," adding that her and Carson are "so in love.”
Alex, who won Richie Strahan's heart on season four of The Bachelor and later appeared on the second season of Bachelor in Paradise, first announced she was expecting her second baby earlier this year.
"I wonder if you’ll have your Father’s eyes? Will you be stubborn like your Mother? See you soon little darling," she penned, along with a stunning black and white image of her bump.
Alex kept her fans updated on her second pregnancy, admitting that it started out "entirely different" to her first.
"Naive little me assumed that this journey would mirror my first. My entire pregnancy with Elijah was a breeze!" she wrote.
"I’ll be honest, I was finding it incredibly hard to enjoy the beginning of my pregnancy and at times it got me down," she added.
Alex was memorably left heartbroken after her relationship with Richie Strahan ended after her appearance on The Bachelor in 2016.
The star then made headlines after announcing she was engaged to her girlfriend Maegan Luxa in 2017 before sadly splitting a short time after. She also starred on the 2019 series of Bachelor In Paradise, where she enjoyed a brief romance with Bill Goldsmith.