  Via probabilistic convex integration, we prove non-uniqueness in law of the
two-dimensional surface quasi-geostrophic equations forced by random noise of
additive type. In its proof we work on the equation of the momentum rather than
the temperature, which is new in the study of the stochastic surface
quasi-geostrophic equations. We also generalize the classical
Calder$\acute{\mathrm{o}}$n commutator estimate to the case of fractional
Laplacians.
