A settlement was reached Jan. 10 between the developer of Quincy Center and the owner of a downtown office building, ending a seven-month conflict. Quince Limited Partnership, owner of the four-story office building at 1515 Hancock St., alleged last June that developer Hancock Adams Associates LLC had breached a purchase-and-sale contract. Members of Street-Works Development LLC, the partner in the $1.6 billion downtown redevelopment, own Hancock Adams. According to Jeffrey Levien, senior director of development for Street-Works, Hancock Adams is moving forward to purchase Quince’s property as if there had never been a lawsuit, and a closing on the $12.9 million property is expected by this fall. “It’s a shame litigation was commenced, but in the end, cooler heads prevailed to arrive at an appropriate resolution,” Levien said in an e-mail.