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Goldstein pg 59
It can be shown that if a cyclic coordinate $q_{j}$ is such that $d q_{j}$ corresponds to a rotation of the system of particles around some axis, then the conservation of its conjugate momentum corresponds to conservation of an angular momentum.
Suppose we have a two-body central force problem and reduce it to a single body problem. Then can we use the above quoted text to prove that the angular momentum is conserved?
While the quoted text above is correct and gives a worded explanation on how angular momentum is conserved in a two-body central force problem, generally, more rigor is required for a proof. We know that given the Lagrangian of the system the generalized momentum is defined as: $$ p_j = \frac{\partial{L}}{\partial{\dot{q_j}}} $$ and also the Lagrange equations of motion are expressed by: $$ \dot{p}_j=\frac{\partial{L}}{\partial{q_j}} $$ The Lagrangian for a two-body system (reduced to a one-body problem) of reduced mass $\mu$ moving in a central-force field described by the potential function $U(r)$ in polar coordinates is: $$ L=\frac{1}{2}\mu(\dot{r}^2+r^2\dot{\theta}^2)-U(r) $$ from here we can take our angular momentum conjugate to $\theta$ as $$ \dot{p}_\theta = \frac{\partial L}{\partial \theta}=0=\frac{d}{dt}\frac{\partial{L}}{\partial{\dot{\theta}}} $$ since the Lagrange equation of motion in $\theta$ is $$ \frac{\partial L}{\partial \theta}-\frac{d}{dt}\frac{\partial L}{\partial \dot{\theta}}=0 $$ therefore $$ \frac{d}{dt}\frac{\partial L}{\partial \dot{\theta}}=\frac{d}{dt}(p_\theta)=\texttt{constant} $$ Hence, angular momentum is conserved. And this all relies on the fact that $\theta$ is a cyclic coordinate which made its conjugate momentum zero as the quoted text states.
