The Boston Q.Z. Map is a map of the north end side of the Boston quarantine zone in Massachusetts developed by the Federal Disaster Response Agency (FEDRA). It appears to have been labeled by Joel displaying military patrol routes, smuggling tunnels, and locations of other smugglers including Tess, Robert, and Donovan.
The Boston Q.Z. Map is found during the second chapter, "The Quarantine Zone", in the section "20 Years Later". Once you pick up Joel's backpack upon reaching the underground tunnel, it will already be among his personal possessions.
A map of the Boston Quarantine Zone