The pair, who have been together since 2016, both shared the news on Instagram with the same picture of a diamond ring.
Turner, who plays Sansa Stark in fantasy TV drama Game of Thrones, posted the photo with the caption "I said yes".
Jonas's brother Nick, who was also in American pop band The Jonas Brothers, tweeted his congratulations.
He said: "Ahh! Congratulations to my brother... and sister in law to be on your engagement. I love you both so much."
