This is an interesting question. Jack gives a hint to the solution, but let me fill in the details.
What we want to show is that how $R(X,Y,X,Y)$ is related to the second fundamental form.The idea is regard the surface $M$ as a submanifold of $\mathbb R^3$, note that here the metric $g$ of $M$ is induced by the canonical metric of $\mathbb R^3$.
Now, the Levi-Civita connection $\nabla$ on $(M,g)$ satisfies $$(\nabla_XY)(p)=(\nabla_X^EY)(p)^{\top}$$
where $(\nabla^E)$ is the Levi-Civita connection on $\mathbb R^3$, and $v^{\top}$ means projection onto the tangent space of $M$. Let's introduce Gauss's equation in the following theorem:
Gauss Theorem: The curvature tensor $R$ of a submanifold $M\subset \mathbb R^n$ is given by the Gauss equation $$\langle R(X,Y)W,Z\rangle=\alpha(X,Z)\cdot \alpha(Y,W)-\alpha(X,W)\cdot \alpha(Y,Z)$$
Where $\alpha(X,Y)=(\nabla_X^EY)(p)-(\nabla_X^EY)(p)^{\top}=(\nabla_X^EY)(p)^{\bot}$, i.e., projecting vector onto normal space, and $(\nabla^E)$ is the canonical Levi-Civita connection on $\mathbb R^n$. One can see [this reference] 1 for details in section 1.8~1.10.
Specifically, in our case, $M$ is a two dimensional surface in $\mathbb R^3,$ where we have the parameterization $r=r(u,v)$, with spanning vector fields of tangent bundle $\{r_u, r_v\}$. Also in $\mathbb R^3$, the normal space of $M$ at a point is spanned by a unit normal vector $n$, and we can write $\alpha(X,Y)=\nabla_X^EY\cdot n$. And by Gauss theorem, we have $$-\langle R(r_u,r_v)r_u,r_v\rangle=\alpha(r_u,r_u)\cdot \alpha(r_v,r_v)-\alpha(r_u,r_v)\cdot \alpha(r_v,r_u)$$
Now the second fundamental form is given by $L=r_{uu}\cdot n, M= r_{uv}\cdot n$, and $N=r_{vv}\cdot n$ (note here I use different notation of second fundamental form as yours).
$$\alpha(r_u,r_u)=\nabla_{r_u}^Er_u\cdot n=\frac{\partial}{\partial u}r_u\cdot n=r_{uu}\cdot n=L,$$
and the others expressions are similar, thus we have
$$-\langle R(r_u,r_v)r_u,r_v\rangle=LM-N^2$$
which gives us the second fundamental form as we desired.
Best Answer
Good question. These results are surprisingly hard to find. (Books by Helgason and by Eberlein, which are standard references for geometry of symmetric spaces, do not contain the results.)
Chavel, Isaac, On Riemannian symmetric spaces of rank one, Adv. Math. 4, 236-263 (1970). ZBL0199.56403.
Heintze, Ernst, On homogeneous manifolds of negative curvature, Math. Ann. 211, 23-34 (1974). ZBL0273.53042.
Most likely, one can find even earlier references (probably by sifting through papers by Elie Cartan.)