"Nico" is not a Czech word. You could add "nic" if you needed: "Nemáme nic co jíst." but it is not to natural, it is just not needed there. The "co" is not here instead of "nic" and it is not here as the translation of "nothing". The "co" comes from the phrase "mít co jíst" "to have something to eat".

A good natural sentence with "nic" is "Nemáme nic k jídlu.".