  We argue that only exact, comprehensive, and explicit solutions for the
fundamental models, the Klein-Gordon (KG) oscillators and the KG-Coulomb, would
help to understand the effects of gravitational fields on the dynamics of such
quantum mechanical systems. In the current methodical proposal, the effects of
the gravitational fields generated by a G\"odel-type Som-Raychaudhuri (SR)
cosmic string spacetime on KG-oscillators (KG-particles in general) are studied
and reported. In so doing, we revisit the KG-oscillators in a topologically
trivial G\"odel-type spacetime background and use textbook procedures to report
its exact solution that covers even and odd parities. Next, we discuss the
drawbacks associated with the power series expansion approach that implies the
biconfluent Heun functions/polynomials solution. We, therefore, recollect the
so called pseudo perturbative shifted L expansion technique (PSLET) as an
alternative and more sophisticated method/technique. Illustrative examples are
used: (i) a KG-oscillator in a topologically trivial G\"odel-type spacetime,
(ii) a quasi-free KG-oscillator in G\"odel SR-type cosmic string spacetime,
(iii) a KG-Coulombic particle in G\"odel SR-type cosmic string spacetime at
zero vorticity, and (iv) a massless KG-particle in G\"odel SR-type cosmic
string spacetime in a Cornell-type Lorentz scalar potential. The corresponding
exact energies are obtained from the zeroth (leading) order correction of
PSLET, where all higher order correction identically vanish. The comprehensive
exactness of the reported solutions manifestly suggest degeneracies associated
with spacetime (STAD) phenomenon.
