Heat oil in a large skillet over medium heat. Add onion and garlic; cook and stir until the onion has softened and turned translucent, about 5 minutes.
Stir in ginger and curry paste; cook and stir for 1 minute.
Pour in coconut milk; bring to a simmer. Stir in fish sauce and brown sugar.
Add bell peppers and zucchini; simmer until vegetables are tender, about 10 minutes.