Hydrometeorological time series are inherently stochastic and exhibit temporal dependence, commonly modeled using Gaussian autoregressive moving average (ARMA) models. However, the normality assumption is often too restrictive for environmental variables such as streamflow, which are nonnegative and right-skewed. We propose the Wei-ARMA model, a new class of ARMA models based on the Weibull distribution that incorporates ARMA components, external regressors, and a link function. A parametric trend test is also introduced, with parameters estimated via the conditional maximum likelihood method. Monte Carlo simulations assess finite-sample performance. An application to streamflow data from the Vacacaí River, Brazil, shows that the proposed model captures key statistical features, avoids unrealistic negative predictions, and outperforms the Gaussian ARMA. Mean absolute percentage errors in in-sample prediction are reduced by 23%, 34%, and 9% for mean, maximum, and minimum monthly streamflow, respectively, relative to the Gaussian ARMA model. The proposed trend test successfully detects monotonic trends.