The early-stage Alzheimer's disease (AD) detection has been considered an important field of medical studies. Like traditional machine learning methods, speech-based automatic detection also suffers from data privacy risks because the data of specific patients are exclusive to each medical institution. A common practice is to use federated learning to protect the patients' data privacy. However, its distributed learning process also causes performance reduction. To alleviate this problem while protecting user privacy, we propose a federated contrastive pre-training (FedCPC) performed before federated training for AD speech detection, which can learn a better representation from raw data and enables different clients to share data in the pre-training and training stages. Experimental results demonstrate that the proposed methods can achieve satisfactory performance while preserving data privacy.