In our investigation, we explore the quantum dynamics of charge-free scalar particles through the Klein-Gordon equation within the framework of rainbow gravity's, considering the Bonnor-Melvin-Lambda (BML) space-time background. The BML solution is characterized by the magnetic field strength along the axis of symmetry direction which is related with the cosmological constant Lambda and the topological parameter alpha of the geometry. The behavior of charge-free scalar particles described by the Klein-Gordon equation is investigated, utilizing two sets of rainbow functions: (i) f(chi)=(e(beta chi-1))/beta chi, h(chi)=1 and (ii) f(chi)=1, h(chi)=1+beta(chi)/2. Here 0<(chi=(|E|)/E-p)<= 1 with E represents the particle's energy, E-p is the Planck's energy, and beta is the rainbow parameter. We obtain the approximate analytical solutions for the scalar particles and conduct a thorough analysis of the obtained results. Afterwards, we study the quantum dynamics of quantum oscillator fields within this BML space-time, employing the Klein-Gordon oscillator. Here also, we choose the same sets of rainbow functions and obtained approximate eigenvalue solution for the oscillator fields. Notably, we demonstrate that the relativistic approximate energy profiles of charge-free scalar particles and oscillator fields get influenced by the topology of the geometry and the cosmological constant. Furthermore, we show that the energy profiles of scalar particles get modifications by the rainbow parameter and the quantum oscillator fields by both the rainbow parameter and the frequency of oscillation
