whenever i'm in that lunch break situation with a female coworker this conversation always happens:
her: "So what days do you usually have off?"
me: "uh, Mondays and Wednesdays."
her: "what do you do when you're off?"
me: "umm, just relax and walk to this nearby thai restaurant and drink boba locas."
*long pause while I battle conversation anxiety*
me: "so what about you, what do you do on your days off?"
her: "go to clubs, bars, drink with my (millions) of friends and wake up in random ppls' houses."
me: "what kind of music do they play at the clubs you go to?"
her: "oh everything, sometimes dubstep blah blah..."
then I plateau socially, as both of us decide that we're just BS'ing and that she's not actually interested in inviting me to go clubbing with her and her cool friends due to my general awkwardness. but that's fine, I guess. at least i'm able (barely) to get through a conversation with a girl even if I have no chance at scoring a date with her.
anyway, the point i'm tryin to make is you should try to get a conversation going with this guy next time you're sitting together even if the whole process bears no fruit. just figure out a way to talk about music or movies. a good opener line is
"have you seen Wolverine(or whatever movie you saw lately or want to see)?" then work from there