No internal and private function can't be declared as payable
you will get a compiler error
TypeError: "internal" and "private" functions cannot be payable
The idea of having payable modifier is "to permission or prevent intentional and accidental value transfers" and using it twice first in external/public function and then again in internal function which is called by that function does not make sense :)
but you can use msg.value inside the internal function
so something like this
function addValue() external payable {
_addValue();
}
function _addValue() internal {
payable(addr).transfer(msg.value);
}