The property is in front of my apartment complex's manaement office - since it was the grass between the street curb and the sidewalk I am not sure if that makes it the property of NYC.
Should I inform my landlord of the hole and let them know I fell?
Yes. But I would recommend you speak with a personal injury attorney immediately. If you would like you may feel free to call my office, regularly handle these types of cases in Queens and the rest of New York. Our number is (516) 280-4716.