Base Quest are unlocked when you capture a base, "ALL ZOMBIES MUST DIE" will appear in the center of the screen. There are a total of three in the game. Recommend you attempt this with at least one Smart Bomb, and having at least a shotgun with full ammo. This will be easier done earlier during the game.
Walk into the base that is under zombie control. You will get a message that will say "want to make this area safe ?", select yes and start killing all zombies. If you get hit while doing this,let the zombies kill you, you will respawn in the area again. Best thing to do is to run around and have the zombies follow you, once they get into groups start to shoot them. Towards the last dozen zombies will be SWAT zombies, this is where you will want to use your Smart Bomb. Run around in a circle, that way they are all bunched up next to each other. Once all the SWAT zombies are in a tight space, release the Smart Bombs to kill them all.