Income approach will sum all gross profits, rents, interest, wages and indirect taxes minus subsidies across the economy. If you want to separate GDP by industry then for public administration you just sum all of the above for its category.
Public administration includes among other things:
Services like transportation, maintaining law and order, road construction, police, jails, tax exemptions, medicare, social security,
Some of the above won't generate profit but some can (for example public transportation). The same holds for interest, rents, wages and so on. For GDP in public administration you just sum gross incomes in this category.