As other answers have mentioned, this almost certainly a scam.
That said, the answer to the actual political question of whether the royal family needs permission to call people on the phone is probably not. For instance, here Princes William and Harry talk about a phone call they had with their mother, Princess Diana. While we can't say for sure that protocol hasn't changed, or that they didn't have to ask permission, there's no mention of it.
Further, a description of how a member of the royal family tended to receive phone calls, while undoubtedly more formal than how the average person does it, implies that she didn't have to ask anyone's permission:
He said: “That officer would ask some questions and then pass the call
on to the princess’s dresser, who would know absolutely everything
there was to know about the princess and her acquaintances, and if
they were satisfied they would ask the princess if they wanted to take
Prince Phillip apparently also made phone calls.
As additional general supporting evidence, this Q&A thread on Quora, the members of the royal family use smartphones, and there are various pictures of them using them casually. While this doesn't prove that they only use hangouts without permission, it does suggests that their communications aren't being tightly controlled.
The top answer in that thread also claims that Prince William called the Queen when he had a child. Again, perhaps she'd given him special permission, and perhaps he used Hangouts, but it doesn't seem likely.
Finally, the royal family of Britain is largely composed of adults, many of whom spend substantial time away from Britain, to say nothing of their residence. It would seem completely impracticable for a family member abroad to get permission every time they needed to make a call. Especially before the advent of smartphones: how would they get the permission necessary to make a call?
None of this confirms for certain that members of the royal family don't need to ask permission (from whom? the Queen) to make phone calls, but it does make it seem unlikely.