Apply for Japanese citizenship

We are mexican couple residing in Japan almost 1 year under student visa category.But our child Born in Japan.Can we apply for Japanese passport and Japanese citizenship for our child. My husband also
mexican citizen.

No

Citizenship in Japan is by blood - not by place of birth. Your son is Mexican. It doesn't matter where he's born - if you or your husband are not Japanese, then neither is he.

There are three ways a person may obtain Japanese citizenship by birth.

1) if the father or mother is a Japanese citizen at the time of birth
2) if the father died before the child�fs birth, but was a Japanese citizen
3) if the child is born in Japan and both of the parents are unknown or without nationality.

In your case the answer is no

Just don't forget to register your child at the Mexican embassy (it is important) and at the city office. The child will be registred as a foreign resident.
No Japanese passport, no koseki for your child. If you thought Japan was like Amurika, you definitely thought wrong. Hard to imagine you spent 1 year here and did not notice that the place was not really foreign friendly...

Japan is a lot more foreigner-friendly than many similar countries, as long as you know its language and respect its rules.