You can plant on your nature strip without Council approval if you meet the guidelines. However, you will need approval from Council and also VicRoads if your property is on a main arterial road.

Native plants only – Use drought tolerant indigenous ground covers, shrubs and grasses available for purchase from Bayside Community Nursery or other indigenous plant nursery

Bayside, includes Brighton, Hampton, Black Rock and Beaumaris

See https://www.bayside.vic.gov.au/how-plant-your-nature-strip for guidelines, plant lists etc.