We present the first search for the Majoron-emitting modes of the neutrinoless double $β$ decay ($0νββχ_0$) using scintillating cryogenic calorimeters. We analysed the CUPID-0 Phase I data using a Bayesian approach to reconstruct the background sources activities, and evaluate the potential contribution of the $^{82}$Se $0νββχ_0$. We considered several possible theoretical models which predict the existence of a Majoron-like boson coupling to the neutrino. The energy spectra arising from the emission of such bosons in the neutrinoless double $β$ decay have spectral indices $n=$ 1, 2, 3 or 7. We found no evidence of any of these decay modes, setting a lower limit (90% of credibility interval) on the half-life of 1.2 $\times$ 10$^{23}$ yr in the case of $n=$ 1, 3.8 $\times$ 10$^{22}$ yr for $n=$ 2, 1.4 $\times$ 10$^{22}$ yr for $n=$ 3 and 2.2 $\times$ 10$^{21}$ yr for $n=$ 7. These are the best limits on the $0νββχ_0$ half-life of the $^{82}$Se, and demonstrate the potentiality of the CUPID-0 technology in this field.