Static deep neural network (DNN) watermarking techniques typically employ irreversible methods to embed watermarks into the DNN model weights. However, this approach causes permanent damage to the watermarked model and fails to meet the requirements of integrity authentication. Reversible data hiding (RDH) methods offer a potential solution, but existing approaches suffer from weaknesses in terms of usability, capacity, and fidelity, hindering their practical adoption. In this paper, we propose a novel RDH-based static DNN watermarking scheme using quantization index modulation (QIM). Our scheme incorporates a novel approach based on a one-dimensional quantizer for watermark embedding. Furthermore, we design two schemes to address the challenges of integrity protection and legitimate authentication for DNNs. Through simulation results on training loss and classification accuracy, we demonstrate the feasibility and effectiveness of our proposed schemes, highlighting their superior adaptability compared to existing methods.