This paper presents the first concolic testing framework explicitly designed for quantum programs. The framework introduces quantum constraint generation methods for quantum control statements that quantify quantum states and offers a symbolization method for quantum variables. Based on this framework, we generate path constraints for each concrete execution path of a quantum program. These constraints guide the exploration of new paths, with a quantum constraint solver determining outcomes to create novel input samples, thereby enhancing branch coverage. Our framework has been implemented in Python and integrated with Qiskit for practical evaluation. Experimental results show that our concolic testing framework improves branch coverage, generates high-quality quantum input samples, and detects bugs, demonstrating its effectiveness and efficiency in quantum programming and bug detection. Regarding branch coverage, our framework achieves more than 74.27% on quantum programs with under 5 qubits.