  We present here a quantum cosmological model with Bohm-de Broglie
interpretation of the theory described by a combination of two terms of the Fab
Four cosmological theory. The first term is the John Lagrangian and the second
is a potential representing matter content to avoid classical trivial
solutions. This model has two free functions that provide an adjustment
mechanism known classically as self-tuning. The self-tuning is a way to address
the cosmological constant problem by allowing a partial break of symmetry in
the scalar field sector. The Fab Four is the most general set of self-tuning
scalar-tensor gravitational theories in four dimensions. The minisuperspace
Hamiltonian thus obtained from this combination of Fab Four terms has
fractional powers in the momenta, leading to a problem in applying canonical
quantization. We have solved this problem by generalizing the canonical
quantization rule using the so-called conformable fractional derivative. We
show that this analysis leads to both singular and bouncing (non-singular)
solutions, depending on the initial conditions over the scale factor and the
homogeneous scalar field, and also depending on the free functions mentioned.
This provides an adjustment mechanism in analogy with the classical self-tuning
of the Fab Four, but with another interpretation.
