Olivia Dunham prosecuted Broyles' good friend Sanford Harris for sexually assaulting three women. This led to a strong antipathy from Broyles towards Olivia, which was overcome only by Agent Dunham's overwhelming competence.
The two are introduced when Flight 627 lands in Boston. Initially, there was some antagonism between the two due to their background. In addition, Broylees showed little faith that Olivia can conduct an investigation properly.
Broyles was later shocked when Olivia later prompted the him to help her secure the release of Walter Bishop, an act that required jumping through alot of loopholes. Broyles was impressed by the amount of work she had competently done so he was willing to forget and start fresh with her. The events with the events of Flight 627 convinced that Broyles can trust Olivia. Broyles dares to trust Olivia more. He helps Olivia get Peter returned from Baghdad despite the massive potential trouble he could be in. In addition, he finds confience in Olivia that she has successfully resolved the strange disease that plagued the plane, that he decides to take a farther leap. Broyles introduces Olivia to the Fringe Program and enlists her to participate. There, she is sworn to secrecy, having given her a huge security clearance - not to mention promotion - for her to aid in the search for the "Pattern". Having elisted her, Broyles assigns her to the people she worked with, Peter, Walter, and Astrid. ("Pilot")
The professional relationship has remained steady. Despite being on the same team, Broyles kept some information from Olivia, especially on a case she is working on. ("The Ghost Network ")
By the third season,the relationship has become a firm friendship and enormous trust between the two. It is evident that Broyles had a bit of a hard time beleiving that Faulivia is from another universe when they switched places.
Unfortunately this relationship ends when Broyles in the final episode of the season dies.
Saddened, Olivia is discontent that she has to deal with a alternate Broyles.