The Generalized Uncertainty Principle (GUP) represents a fundamental modification to Heisenberg's uncertainty principle, incorporating quantum gravitational effects through the introduction of both minimal and maximal length scales. In this work, we investigate the cosmological implications of GUP by examining its effects on the solutions of the Friedmann equations across different evolutionary eras of the universe. We employ the Hamiltonian formalism to derive modified Friedmann equations that incorporate GUP corrections. The Hamilton procedure provides a systematic framework for obtaining these quantum-corrected gravitational equations while preserving the consistency of the theoretical structure. To solve these modified equations, we apply the classical perturbation method, treating GUP corrections as small perturbations to the standard cosmological solutions. This approach allows us to obtain analytical expressions for the modified cosmological parameters while maintaining tractability. Our analysis spans multiple epochs of cosmic evolution, including the radiation-dominated, matter-dominated, and dark energy-dominated eras. For each epoch, we derive the corresponding modified solutions and analyze how GUP-induced corrections alter the universe's expansion dynamics. We place particular emphasis on the Cosmic Microwave Background (CMB) era, which provides robust observational constraints on cosmological models. Using our GUP-modified framework, we compute the corrected value of the Hubble constant and perform a detailed comparison with the observationally determined value from CMB measurements. This comparison serves as a critical test for the viability of GUP modifications in cosmology and enables us to constrain the phenomenological parameters associated with minimal and maximal length scales. Our results contribute to understanding how quantum gravitational effects, as encoded in the GUP framework, might influence large-scale cosmological observables and potentially address tensions in current cosmological measurements.