We present a quantum cosmological theory in which the observed late-time acceleration of the Universe emerges from the discrete quantum structure of spacetime, without requiring a dark energy component. The theory is built on four postulates: (1) the Universe's volume is quantized in Planck units, Vn=nV0Vn=nV0 with n∈Nn∈N; (2) cosmic expansion proceeds via discrete transitions n→n+1n→n+1; (3) transition rates scale as Γn=Γ0n⋅n/(n+nc)Γn=Γ0n⋅n/(n+nc); (4) physical time is derived from the sequence of transitions. These assumptions yield a scale-dependent Hubble parameter H(a)=H0Ωm,0a−3+Ωr,0a−4⋅a−3ϵ(a)H(a)=H0Ωm,0a−3+Ωr,0a−4⋅a−3ϵ(a) with ϵ(a)=−12a3/(a3+ac3)ϵ(a)=−21a3/(a3+ac3). The effective equation of state transitions from weff≈0weff≈0 at early times to weff→−1weff→−1 at late times, matching observations. The single free parameter ncnc corresponds to a transition redshift zc≈0.6zc≈0.6, naturally explaining the onset of acceleration. The theory is falsifiable through precise measurements of Type Ia supernovae, baryon acoustic oscillations, cosmic microwave background anisotropies, and the growth of large-scale structure.