  We use two correlated metric functions and transform/deform the pointlike
global monopole (PGM) spacetime metric into a quasi-PGM (QPGM) spacetime one.
We study Klein-Gordon (KG) particles (manifestly introduced by the non-minimal
coupling form of the operator $% \tilde{D}_{\mu }=D_{\mu }+\mathcal{F}_{\mu }$,
with $\mathcal{F}_{\mu }$ $% \in %TCIMACRO{\U{211d} }% %BeginExpansion
\mathbb{R} %EndExpansion $, and $D_{\mu }=\partial _{\mu }-ieA_{\mu }$) in such
QPGM spacetime background. We show that the KG-particles in the QPGM spacetime
are isospectral and invariant with KG-particles in the PGM spacetime, provided
that the two metric functions are correlated. The Wu-Yang magnetic monopole
(WYMM) is also included in the process. We report the effects of both monopoles
PGM/QPGM and WYMM on the spectroscopic structure of two models, KG-oscillators
and KG-Coulomb particles in such spacetime background. Moreover, we discuss
KG-Coulomb particles and shifted KG-oscillators in a PGM spacetime and a WYMM,
where such models are manifestly introduced by some Lorentz scalar interaction
potentials $S(r)$. The current study is not only of fundamental observability
nature but also of pedagogical interest in quantum gravity.
