The analysis of the Chandra X-ray observations of the gravitationally lensed quasar RX J1131-1231 revealed the detection of multiple and energy-variable spectral peaks. The spectral variability is thought to result from the microlensing of the Fe K-alpha emission, selectively amplifying the emission from certain regions of the accretion disk with certain effective frequency shifts of the Fe K-alpha line emission. In this paper, we combine detailed simulations of the emission of Fe K-alpha photons from the accretion disk of a Kerr black hole with calculations of the effect of gravitational microlensing on the observed energy spectra. The simulations show that microlensing can indeed produce multiply peaked energy spectra. We explore the dependence of the spectral characteristics on black hole spin, accretion disk inclination, corona height, and microlensing amplification factor, and show that the measurements can be used to constrain these parameters. We find that the range of observed spectral peak energies of QSO RX J1131-1231 can only be reproduced for black hole inclinations exceeding 70 degree and for lamppost corona heights of less than 30 gravitational radii above the black hole. We conclude by emphasizing the scientific potential of studies of the microlensed Fe K$alpha$ quasar emission and the need for more detailed modeling that explores how the results change for more realistic accretion disk and corona geometries and microlensing magnification patterns. A full analysis should furthermore model the signal-to-noise ratio of the observations and the resulting detection biases.
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