<p>The motion of a point on an ellipsoid can be interpreted using the generation of elliptical rotations. An elliptical rotation can be defined using elliptic quaternions. The Fibonacci sequence, defined by the second-order linear relation, is a famous sequence used in mathematics, physics, computer science, technical analysis and others. In this paper, we introduce and study generalized Fibonacci-Leonardo elliptic quaternions, which generalize the Fibonacci elliptic quaternions and Leonardo elliptic quaternions, simultaneously. We give the ordinary generating functions, the Binet-type formulas, general bilinear index-reduction formulas, and the sum of the finite, finite odd, and finite even terms of these quaternions.</p>