Sultanpur National Park is the classic birding trip from Gurgaon, about an hour west of the corporate districts if traffic behaves. The park is small, centred on a lake and wetland habitat, but it punches above its size in winter when migratory birds arrive.
Come early, bring binoculars, and keep expectations seasonal. In the dry heat it can feel sparse; after monsoon and through winter it is far more rewarding. It also combines well with Farrukhnagar if you want a heritage add-on outside the city.