Quantum sensing technologies have experienced rapid progresses since entering the `second quantum revolution'. Among various candidates, schemes relying on Rydberg atoms exhibit compelling advantages for detecting radio frequency signals. Based on this, Rydberg atomic quantum receivers (RAQRs) have emerged as a promising solution to classical wireless communication and sensing. To harness the advantages and exploit the potential of RAQRs in wireless sensing, we investigate the realization of the direction of arrival (DOA) estimation by RAQRs. Specifically, we first conceive a Rydberg atomic quantum uniform linear array (RAQ-ULA) aided wireless receiver for multi-target DOA detection and propose the corresponding signal model of this sensing system. Our model reveals that the presence of the radio-frequency local oscillator in the RAQ-ULA creates sensor gain mismatches, which degrade the DOA estimation significantly by employing the classical Estimation of Signal Parameters via Rotational Invariant Techniques (ESPRIT). To solve this sensor gain mismatch problem, we propose the Rydberg atomic quantum ESPRIT (RAQ-ESPRIT) relying on our model. Lastly, we characterize our scheme through numerical simulations, where the results exhibit that it is capable of reducing the estimation error of its classical counterpart on the order of $> 400$-fold and $> 9000$-fold in the PSL and SQL, respectively.