No, there is no built-in option to secure a custom page via Tridion itself (i.e. to its users and groups). Some might say that this makes sense, as a custom page is really just a URL that is loaded in an iframe, so if you know the URL then you can probably by-pass Tridion anyway.
Of course, if you're hosting your custom page within the same system/environment as the Tridion web site then you can probably implement the same authentication system (integrated authentication, SSO etc, whatever) as a basic check. You could perhaps even check whether the user is a member of a specific group in Tridion, but it would still essentially be custom code running within the context of your custom page.