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 sqrt(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 x 10^-5 for Z to pi^0gamma, 1.46 x 10^-5 for Z to gammagamma, and 1.52 x 10^-5 for Z to pi^0pi^0. The Z to pi^0gamma and Z to gammagamma limits improve by factors of 2.6 and 3.6, respectively, the most stringent results from other experiments. The Z to pi^0pi^0 branching ratio limit is the first experimental result on this decay.
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