In this paper (arXiv:2412.06057) we revisit the nonlinear second-order differential equations x´´(t)=a(x)x´(t)^2+b(t)x´(t), where a(x) and b(t) are arbitrary functions on their argument from the perspective of Lie-Hamilton systems. For the particular choice a(x)=3/x and b(t)=1/t, these equations reduce to the Buchdahl equation considered in the context of General Relativity. It is shown that these equations are associated to the ‘book’ Lie algebra 𝔟2, determining a Lie-Hamilton system for which the corresponding t-dependent Hamiltonian and the general solution of the equations are given. The procedure is illustrated considering several particular cases. We also make use of the quantum deformation of 𝔟2 with quantum deformation parameter z (where q=e^z), leading to a deformed generalized Buchdahl equation. Applying the formalism of Poisson-Hopf deformations of Lie-Hamilton systems, we derive the corresponding deformed t-dependent Hamiltonian, as well as its general solution. The presence of the quantum deformation parameter z is interpreted as the introduction of an integrable perturbation of the initial generalized Buchdahl equation, which is described in detail in its linear approximation.