We study the effect of a single non-magnetic impurity in A$_{y}$Fe$_{2-x}$Se$_{2}$ (A=K, Rb, or Cs) superconductors by considering various pairing states based on a three-orbital model consistent with the photoemission experiments. The local density of states on and near the impurity site has been calculated by solving the Bogoliubov-de Gennes equations self-consistently. The impurity-induced in-gap bound states are found only for attractive impurity scattering potential, as in the cases of doping of Co or Ni, which is characterized by the strong particle-hole asymmetry, in the nodeless $d_{x^2-y^2}$ wave pairing state. This property may be used to probe the pairing symmetry of FeSe-based 122-type superconductors.
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