Precise positioning is becoming increasingly crucial across various applications, including rescue operations, intelligent transportation, logistics, and environmental monitoring. However, previous research often assumes that wireless sensor networks are perfectly synchronized or have unlimited communication resources. These assumptions frequently do not align with real-world scenarios, particularly in resource-constrained networks with stringent power and communication limits. In this paper, we propose a target localization and time synchronization algorithm based on the quantization of time-of-arrival (TOA) measurements. This approach accounts for both the quantization process and clock offsets caused by asynchronous clocks between sensor nodes and the target source. To tackle the problem, a variational Bayesian learning method is proposed to estimate the target source location and clock offsets jointly. This algorithm aims to find a feasible variational distribution that approximates the true posterior distribution, thereby conducting the Bayesian estimation over the approximate distributions and obtaining the accurate locations and clock offsets. Additionally, we derive the quantized Bayesian Cramr-Rao Bound (QBCRB) for quantized TOA measurements to evaluate localization and time synchronization performance. Simulation and experimental results demonstrate that the proposed method provides effective estimations and outperforms other comparison algorithms across various scenarios.