We got an apartment on lease for 3 month initial lease agreement. The lease is suppose to convert to 1 month beyond the initial agreement period. Now if we want to vacate the apartment immediately after completion of the initial lease period and not a day more.. are we entitled for the 30 days notice and rent?

What does the lease say? Normally a lease especially one with an automatic renewal like that will specify how much notice has to be given to continue the lease.