Avian blood analysis is a fundamental method for investigating a wide range of topics concerning individual birds and populations of birds. Determining precise blood cell counts helps researchers gain insights into the health condition of birds. For example, the ratio of heterophils to lymphocytes (H/L ratio) is a well-established index for comparing relative stress load. However, such measurements are currently often obtained manually by human experts. In this article, we present a novel approach to automatically quantify avian red and white blood cells in whole slide images. Our approach is based on two deep neural network models. The first model determines image regions that are suitable for counting blood cells, and the second model is an instance segmentation model that detects the cells in the determined image regions. The region selection model achieves up to 97.3% in terms of F1 score (i.e., the harmonic mean of precision and recall), and the instance segmentation model achieves up to 90.7% in terms of mean average precision. Our approach helps ornithologists acquire hematological data from avian blood smears more precisely and efficiently.