I don't really think there's much point in calling it a "Rivalry" either.
They are at the top of the Beast Pirates hierarchy already, and the only spot that would be left for them to compete would be replacing Kaido, which obviously is not going to happen.
We've seen that they don't have authority over the other, and both were shown to command F6 on various ocasions, such as Queen getting Who's Who and Hawkins to fetch X Drake or dispatching X Drake and Page One to fight Sanji, and King's call for them to target Momonosuke.
They are actually the only one's shown giving direct orders to the F6, unlike Jack who shares the All Star position but never displayed that level of control over Kaido's officers.
On top of that, they even lord over Jack ordering him around and treating him as their lackey
Once again highliting them even within the All Star rank.
They are at the very top and on equal footing, while everyone beneath them answers to them in one way or another and is aware of their superiority.