Finite-temperature correlation functions provide fundamental information about the excitations and response properties of quantum many-body systems. Recently, the quantum minimally entangled typical thermal states (QMETTS) algorithm was introduced for calculating thermal averages of certain observables on near-term quantum devices. However, due to the computational cost of the quantum imaginary time evolution (QITE) subroutine underlying the QMETTS algorithm, the calculation of general thermal quantities with QMETTS remains challenging. Here, we report the calculation of finite-temperature correlation functions of quantum spin models with QMETTS. We describe how to reduce the cost of calculations by exploiting Hamiltonian symmetries and other constraints to eliminate qubits and reduce measurements. Our work advances efforts to study finite-temperature properties of quantum many-body systems on quantum computers.