Two problems of memoryless quantization and data fusion for the detection of a weak signal in stationary dependent noise are addressed: (i) fusion from sensors with mutually independent observations across sensors but dependent across time and (ii) fusion from sensors with correlated observations across time and sensors. For each problem, we consider four distinct schemes (a) fusing the test statistics formed by the sensors without previous quantization (b) quantizing suboptimally each observation and then fusing, (c) quantizing optimally each observation and then fusing, and (d) quantizing optimally each test statistic of the sensors and then fusing the observation sequence of each sensor consists of a common weak signal disturbed by an additive stationary m-dependent, f-mixing or p-mixing noise process. To guarantee high-quality performance, a common large sample size is employed by each sensor. Design criteria are developed from the Neyman-Pearson test in the fusion center for the optimal memoryless sensor test statistics and the sensor quantizer parameters (quantization levels and breakpoints); these design criteria are shown to involve an extension of the asymptotic relative efficiency used in single-sensor detection and quantization. Numerical results in support of the analysis are given for the case of dependent p=mixing Cauchy noise.