Yes, the airport's private owner, Heathrow Airport Limited, will apply to the government's Planning Inspectorate for an unusual kind of national planning consent called a Development Consent Order.
From the airport company's official expansion website:
Expansion will be authorised via a Development Consent Order, a special kind of consent for major infrastructure projects in England and Wales.
https://www.heathrowexpansion.com/the-expansion-plan/planning-process/