A search for forbidden and exotic Z boson decays in the diphoton mass spectrum is presented for the first time in hadron collisions, based on data corresponding to 10.0 fb(-1) of integrated luminosity from proton-antiproton collisions at √s = 1.96 TeV collected by the CDF experiment. No evidence of signal is observed, and 95% credibility level Bayesian upper limits are set on the branching ratios of decays of the Z boson to a photon and neutral pion (which is detected as a photon), a pair of photons, and a pair of neutral pions. The observed branching ratio limits are 2.01 × 10(-5) for Z → π(0)γ, 1.46 × 10(-5) for Z → γγ, and 1.52 × 10(-5) for Z → π(0)π(0). The Z → π(0)γ and Z → γγ limits improve the most stringent results from other experiments by factors of 2.6 and 3.6, respectively. The Z → π(0)π(0) branching ratio limit is the first experimental result on this decay.