Projected Inverse Dynamics Control (PIDC) is commonly used in robots subject to contact, especially in quadrupedal systems. Many methods based on such dynamics have been developed for quadrupedal locomotion tasks, and only a few works studied simple interactions between the robot and environment, such as pressing an E-stop button. To facilitate the interaction requiring exact force control for safety, we propose a novel interaction force control scheme for underactuated quadrupedal systems relying on projection techniques and Quadratic Programming (QP). This algorithm allows the robot to apply a desired interaction force to the environment without using force sensors while satisfying physical constraints and inducing minimal base motion. Unlike previous projection-based methods, the QP design uses two selection matrices in its hierarchical structure, facilitating the decoupling between force and motion control. The proposed algorithm is verified with a quadrupedal robot in a high-fidelity simulator. Compared to the QP designs without the strategy of using two selection matrices and the PIDC method for contact force control, our method provided more accurate contact force tracking performance with minimal base movement, paving the way to approach the exact interaction force control for underactuated quadrupedal systems.