  It is shown that the complex Bernoulli differential equations admitting the
supplementary structure of a Lie-Hamilton system related to the book algebra
$\mathfrak{b}_2$ can always be solved by quadratures, providing an explicit
solution of the equations. In addition, considering the quantum deformation of
Bernoulli equations, their canonical form is obtained and an exact solution by
quadratures is deduced as well. It is further shown that the approximations of
$k^{th}$-order in the deformation parameter from the quantum deformation are
also integrable by quadratures, although an explicit solution cannot be
obtained in general. Finally, the multidimensional quantum deformation of the
book Lie-Hamilton systems is studied, showing that, in contrast to the
multidimensional analogue of the undeformed system, the resulting system is
coupled in a non-trivial form.
