What is Star Sapphire's relationship with Green Lantern?

Was she his ex-girlfriend?

In this iteration, Yes.

This Star Sapphire is the Carol Ferris version and her relationship both as Carol and when inhabited by the Star Sapphire personality is erm...complicated to say the least (as you would expect in comics).

In the DCAU

DC Comics has featured a series of women to bear the name Star Sapphire, each of whom is a mortal woman chosen by the Zamarons, the immortal female counterparts of the Guardians of the Universe, to be their queen and wield the crystal weapon. The Star Sapphires are thus a sort of inversion of the Green Lantern Corps, who are chosen by the also-immortal Guardians.

The second Star Sapphire was Carol Ferris, the sometime girlfriend of Hal Jordan, one of the first Green Lanterns. According to the producers of the series, the animated Star Sapphire is based on Ferris.

^Wikia

Wikipedia has more...a lot more...

Caroline "Carol" Ferris is a fictional character appearing in the DC Comics Universe. She is one of many characters who has used the name Star Sapphire, and was the long-time love interest of Hal Jordan, the Silver Age Green Lantern. In her role as Star Sapphire, Ferris has been active as both a supervillain and, more recently, as a superhero.

As Ferris Aircraft's Vice President, Carol Ferris, the only child of aerospace mogul Carl Ferris and his wife Christine, hired Hal Jordan and quickly found herself attracted to the fearless test pilot, who was secretly the superhero Green Lantern. However, the young couple’s romance quickly became complicated when Carol took over the company from her father and the Zamarons crowned her the new Star Sapphire. When the Zamarons discovered that she was in love with Green Lantern, a servant of their estranged friends, the Guardians of the Universe, they sent her to defeat Green Lantern in battle as Star Sapphire.

Over the years, Star Sapphire and Green Lantern would duel again and again, but each time Jordan would defeat Ferris and revert her to normal. Jordan and Ferris separate when Carol is engaged to Jason Belmore; later she is confined to a wheelchair. It was during this time that Carol found out that Hal Jordan is Green Lantern.

The character/personality (Star Sapphire) has been "inhabited" by several people (in the comics) over the years but has often returned to Carol.

Carol Ferris is first introduced in S.O.S. Green Lantern!, which ran in Showcase vol. 1, issue #22 (October 1959). In her original appearance, Hal Jordan becomes employed at Ferris Aircraft and (after asking her to dinner) she makes it clear that she does not date employees.[2] However, she would go on to play an on and off romantic role in his life. She first appears as the second Star Sapphire in Green Lantern vol. 2, #16 (October 1962). As Star Sapphire, she battles Green Lantern for many years, because the Zamarons want to prove men are inferior. When she is first defeated by him they take away her memory of the event, but the persona keeps resurfacing. When Jordan becomes the Spectre, he removes the Star Sapphire persona from Ferris. While seeking to inhabit the body that Jordan most desires, the Star Sapphire gem again possesses her for a brief period during the Mystery of the Star Sapphire storyline. Her reunion with the Star Sapphire entity is short, however, as it soon learns that Jordan most desires Jillian Pearlman and as a result abandons Ferris.

Though she no longer holds the singular position of Star Sapphire, in Green Lantern vol. 4 issue #38 (March 2009), she receives a violet power ring sent to her by the Zamarons. It attaches itself to her, and she leaves for Zamaron to be inducted as a member of the Star Sapphire Corps. As the Queen dies at the end of Green Lantern (fourth series) #57 (October 2010), Carol is named queen by her predecessor.