On the Gold Coast side, usually not. Cocos palm (Queen palm) is a listed pest species under the City Plan's self-assessable outcome SO5, along with umbrella tree, cadaghi, slash pine and coral tree. The catch is the exemption only applies once an AQF Level 3 arborist identifies and documents it. Casey holds AQF Level 3, so we can do the ID and the removal in one visit, no council application needed.
On the Tweed (NSW) side there's no equivalent pest-species exemption, so a cocos palm there may still need an application if it meets a protected category. We'll tell you which side you're on. More on the council page.