  We describe how to compute the intersection of two Lucas sequences of the
forms $\{U_n(P,\pm 1) \}_{n=0}^{\infty}$ or $\{V_n(P,\pm 1) \}_{n=0}^{\infty}$
with $P\in\mathbb{Z}$ that includes sequences of Fibonacci, Pell, Lucas, and
Lucas-Pell numbers. We prove that such an intersection is finite except for the
case $U_n(1,-1)$ and $U_n(3,1)$ and the case of two $V$-sequences when the
product of their discriminants is a perfect square. Moreover, the intersection
in these cases also forms a Lucas sequence. Our approach relies on solving
homogeneous quadratic Diophantine equations and Thue equations. In particular,
we prove that 0, 1, 2, and 5 are the only numbers that are both Fibonacci and
Pell, and list similar results for many other pairs of Lucas sequences. We
further extend our results to Lucas sequences with arbitrary initial terms.
