Jenna Bush Hager Says 'What About My Children?' When Mom Laura Jokes Barbara's Baby Is 'Favorite'

George W. Bush loved his granddaughter's namesake.

Jenna Bush Hager caught up with her mother, former First Lady Laura Bush, for Thursday's Today show broadcast, sharing their excitement for their family's new addition: Jenna's twin sister Barbara Pierce Bush, 39, gave birth to her first baby, daughter Cora Georgia, with husband Craig Coyne on Monday, Sept. 27.

Mrs. Bush praised the "darling little baby," also sharing how her husband reacted to being a namesake for the newborn.

"It was really great. Cora Georgia — she'll have to be his favorite!" she said with a laugh, as Jenna (who shares Margaret "Mila" Laura, 8, Poppy Louise, 6, and Henry "Hal" Harold, 2, with husband Henry Hager) responded, "No! What about my children? Don't tell him that. He liked the namesake, of course."

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The proud grandparents announced the birth late last month, saying in a statement, "With full hearts, Laura and I are delighted to announce the birth of our new granddaughter. ... Cora is healthy and adorable, and we are proud and grateful."

Barbara's cousin Wendy Wear Stapleton told PEOPLE about the inspiration for the baby name Cora Georgia.

"She likes the alliteration, the Cora Coyne, so I know she was tossing around C names and that was kind of in the lead in late August, so I'm glad she did that," Wear Stapleton said. "Then she's got the Georgia which is for her dad and her grandfather."

George W. Bush/Instagram

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After news of the baby's birth, Jenna celebrated on Instagram, writing: "Dearest Cora Georgia, Today is the day I got to meet my most beautiful, precious, feisty, niece (a bit earlier than we expected!)"

"I witnessed in awe as my dearest, toughest @barbara.p.bush became a mama," she continued alongside a series of photos from the hospital. "And today is the day I fell in love! Some cousins are anxiously waiting to play with you, but not just yet love. Xx auntie j"