Tensile strength, warping degree, and surface roughness are important indicators to evaluate the quality of fused deposition modeling (FDM) parts, and their accurate and stable prediction is helpful to the development of FDM technology. Thus, a quality prediction method of FDM parts based on an optimized deep belief network was proposed. To determine the combination of process parameters that have the greatest influence on the quality of FDM parts, the correlation analysis method was used to screen the key quality factors that affect the quality of FDM parts. Then, we use 10-fold cross-validation and grid search (GS) to determine the optimal hyperparameter combination of the sparse constrained deep belief network (SDBN), propose an adaptive cuckoo search (ACS) algorithm to optimize the weights and biases of the SDBN, and complete the construction of prediction model based on the above work. The results show that compared with DBN, LSTM, RBFNN, and BPNN, the ACS-SDBN model designed in this article can map the complex nonlinear relationship between FDM part quality characteristics and process parameters more effectively, and the CV verification accuracy of the model can reach more than 95.92%. The prediction accuracy can reach more than 96.67%, and the model has higher accuracy and stability.