In recent years, China has proposed the task of promoting technological innovation and improving the quality of development to achieve the innovation-driven development strategy and consolidate the foundation of the real economy. As an important power source and core element to promote the innovation-driven development strategy, scientific and technological talents are conducive to stimulating new economic growth points and releasing the endogenous power of the economy while promoting scientific and technological progress, thus promoting the structural transformation and high-quality economic development of China. The demand forecast for scientific and technological talents is an important basis for the cultivation of scientific and technological talents and policy formulation. The demand for scientific and technological talents has typical non-linear development characteristics and is also influenced by a variety of factors that are difficult to describe quantitatively. Therefore, the grey prediction model with the characteristics of small data and poor information modeling is gradually becoming a common method for forecasting the demand for scientific and technological talents. Although the grey prediction model has the advantages of simple structure and wide applicability, there are still some shortcomings, such as unstable model prediction performance and poor model compatibility. Therefore, starting from the characteristics of the demand for scientific and technological talent data, this article proposes a new grey prediction model FODGM(r,1,kθ,u) with dynamic adaptability based on the defect analysis of the FDGM(1,1,k2) model. The new model adds a nonlinear correction term to the FDGM(1,1,k2) model, which improves the model’s simulation performance of nonlinear original series; the dynamic adaptivity of the model is improved by extending the original first-order accumulation to fractional-order. The new model achieves a better simulation of the nonlinear law of grey information of system development and the totally real number fields expansion of the accumulating order action range, which alleviates the influence of extreme values in the original sequence on the model performance and improves the compatibility of the model structure and the stability of the prediction results. The national research and development (R&D) personnel of full-time equivalent is selected to reflect the demand for scientific and technological talents in China. In order to reflect the robustness of the new model, data of different lengths are reserved for simulation and prediction modeling, and compared with other models. The results show that the FODGM(r,1,kθ,u) model has the smallest comprehensive error in different sample intervals. Finally, the FODGM(r,1,kθ,u) model was applied to predict the future demand for R&D personnel in China. The results showed that the demand for R&D personnel in China is gradually increasing in the future. It is expected that the full-time equivalent of R&D personnel in China will reach 7.29258 million person-years in 2026, indicating significant pressure on the demand side of scientific and technological talents. To this end, relevant countermeasures and suggestions have been proposed. The research results have positive significance in promoting the sustainability of China’s science and technology industry, alleviating the pressure on the demand for scientific and technological talents, and promoting the healthy development of China’s science and technology industry. It has important value for enriching, developing, and improving the theoretical system of grey prediction models.