Yes. Rapper/singer Nicki Minaj does have her own Instagram account. However, because Instagram doesn't have Verified Accounts for celebrities, it's uncertain which account featuring pictures of her is really her official account. The account that appears most likely to be her official account is "@therealnicki_minaj", which states on the top of the page, "Official Instagram for Nicki Minaj." Other Instagram users have commented that her real account may be "@itsnickiminaj".
She has not stated the name of her Instagram account publicly, but she did state recently on Twitter that she has just joined Instagram over the past few months.
On February 14, 2013, she posted pictures on Instagram of herself poolside in a yellow bikini, mostly featuring her backside. Meant in honor of Valentine's Day, the pictures were posted via the Twitter link here.