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→π0γ, 1.46×10−5 for Z→γγ, and 1.52×10−5 for Z→π0π0. The Z→π0γ and Z→γγ limits improve the most stringent results from other experiments by factors of 2.6 and 3.6, respectively. The Z→π0π0 branching ratio limit is the first experimental result on this decay.