We introduce the class of quasi-square-2-isometric operators on a complex separable Hilbert space. This class extends the class of 2-isometric operators due to Agler and Stankus. An operator T is said to be quasi-square-2-isometric if T * 5 T 5 − 2T * 3 T 3 + T * T = 0. In this paper, we give operator matrix representation of quasi-square-2-isometric operator in order to obtain spectral properties of this operator. In particular, we show that the function σ is continuous on the class of all quasi-square-2-isometric operators. Under the hypothesis σ(T) ∩ (−σ(T)) = ∅, we also prove that if E T ({λ}) is the Riesz idempotent for an isolated point of the spectrum of quasi-square-2-isometric operator, then E T ({λ}) is self-adjoint.