"Happiest fourth birthday to our Poppy!" Jenna wrote. "She’s all personality and humor and light. Uniquely Pops—we can’t imagine life without her!!!"
Poppy also shares her special day with Savannah Guthrie's daughter, Vale, who celebrated turning 5 on Tuesday.
While Jenna called her grandfather "Gampy," Poppy was the elder Bush's nickname growing up.
She said on TODAY in May that the former president had shed tears when he heard that she and husband Henry Hager had named their daughter after him.
"He was not expecting it, so he cried,'' Jenna said. "And of course, I cry at like Teacher Appreciation Day commercials (so) I'm, like, crying over the baby and Henry's kind of like wiping the baby's face off because there's like tears in the baby's mouth."
Poppy is Jenna's younger daughter behind Mila, 6, and is now a big sister to the first boy in the family.
"Life with Hal is so, so sweet,'' Jenna wrote on Instagram.