In this paper we analyze quantized spin waves (also referred to as magnons) in bilayers of two-dimensional van der Waals materials, like Vanadium-based dichalcogenides, VX2 (X=S, Se, Te) and other materials of similar symmetry. We assume that the materials exhibit Dzyaloshinski Moriya interaction and in-plane easy-axis magnetic anisotropy due to symmetry breaking induced externally (e.g. by strain, gate voltage, proximity effects to an appropriate substrate/oberlayer, etc.). The considerations are limited to a collinear spin ground state, stabilized by a sufficiently strong in-plane magnetic anisotropy. The theoretical analysis is performed within the general spin wave theory based on the Holstein-Primakoff-Bogolubov transformation. Accordingly, the description takes into account quantum antiferromagnetic fluctuations. However, it is limited to linear spinwave modes. The Dzyaloshinski Moriya interaction is shown to modify the spin wave spectrum of the bilayers, making its low energy part qualitatively similar to the electronic spectrum of the Rashba spin–orbit model.