Here is my theory:
They loved each other around 8 or 9 years ago (remember at the camp when Lucilla talked to Maximus, they both mentioned the kids were around 8 years old?). They almost hit it into a marriage, but Lucilla selected another guy called Verus (actually this guy was Marcus Aurelius co-emperor in the real history) to marry.
This hurt Maximus a lot. That's why in the cell (or on another occasion, can't remember accurately), Lucilla mentioned she hurt Maximus a lot more than he did to her. That's also why Maximus said Lucilla had a insinct for survival (choosing her life based on cool calculation, without emotion involved).
Then they go different paths. Lucilla married Verus, who died early shortly after Lucilla gave birth to Lucius. And Maximus married another dark-haired Spanish girl and also had a son.

Yes, I know is mostly fiction. My question is precisely about Maximus and Lucilla's relationship before the events in the movie. What's your theory?

The movie is mostly fiction, with a shred of truth to it.