This paper presents a novel methodology for binning quasars (QSOs) to identify a "golden" sample suitable for cosmological studies. The authors aim to address the inconsistencies in the Lambda Cold Dark Matter (ΛCDM) model by using QSOs at intermediate redshifts (z = 2.26 to 1100). The method involves a flux-flux relation analysis, selection bias removal, and redshift evolution correction. By applying robust regression techniques, such as the Theil-Sen estimator, the authors minimize the intrinsic scatter of the Ultraviolet (UV) and X-ray fluxes, reducing it to δint = 0.096 ± 0.003. This "gold" sample, composed of 1253 QSOs up to z = 7.54, allows for a more accurate determination of the matter density parameter (ΩM) to be ΩM = 0.240 ± 0.064, aligning with ΛCDM model predictions from Type Ia Supernovae (SNe Ia). The paper details the data selection criteria, the binning procedure, and the robust regression methods used to ensure the reliability of the "gold" sample. The results are validated through cosmological fittings and show independence from initial Hubble constant (H0) values and mild evolution in ΩM. The study also explores a modified gravity framework, specifically f(R)-gravity, to interpret the observed variations in ΩM with redshift.This paper presents a novel methodology for binning quasars (QSOs) to identify a "golden" sample suitable for cosmological studies. The authors aim to address the inconsistencies in the Lambda Cold Dark Matter (ΛCDM) model by using QSOs at intermediate redshifts (z = 2.26 to 1100). The method involves a flux-flux relation analysis, selection bias removal, and redshift evolution correction. By applying robust regression techniques, such as the Theil-Sen estimator, the authors minimize the intrinsic scatter of the Ultraviolet (UV) and X-ray fluxes, reducing it to δint = 0.096 ± 0.003. This "gold" sample, composed of 1253 QSOs up to z = 7.54, allows for a more accurate determination of the matter density parameter (ΩM) to be ΩM = 0.240 ± 0.064, aligning with ΛCDM model predictions from Type Ia Supernovae (SNe Ia). The paper details the data selection criteria, the binning procedure, and the robust regression methods used to ensure the reliability of the "gold" sample. The results are validated through cosmological fittings and show independence from initial Hubble constant (H0) values and mild evolution in ΩM. The study also explores a modified gravity framework, specifically f(R)-gravity, to interpret the observed variations in ΩM with redshift.