Jackson is the 21-year-old daughter of Jade Jagger, whose parents are Mick and Bianca Jagger. Jade is also pregnant and due in June.
"The women in our family just tend to have children young, it runs in the genes," Jackson told Hello magazine.
"I grew up quite quickly and wouldn't have considered having a child if I was doing what my friends are doing at this stage, messing around. I don't drink, I don't smoke and I'm very settled."
Asked about her grandfather's thoughts on her new bundle of joy, Jackson said she imagines "it's nice to be a great-grandad."
"Although I'm not sure he likes the idea of getting old, or being called one," she quipped. "I call him Mick -- I wouldn't start calling him Grandpa."
Jackson and her boyfriend Alex Key, 25, live in Cornwall.