Deep learning-based weather forecasting models have recently demonstrated significant performance improvements over gold-standard physics-based simulation tools. However, these models are vulnerable to adversarial attacks, which raises concerns about their trustworthiness. In this paper, we first investigate the feasibility of applying existing adversarial attack methods to weather forecasting models. We argue that a successful attack should (1) not modify significantly its original inputs, (2) be faithful, i.e., achieve the desired forecast at targeted locations with minimal changes to non-targeted locations, and (3) be geospatio-temporally realistic. However, balancing these criteria is a challenge as existing methods are not designed to preserve the geospatio-temporal dependencies of the original samples. To address this challenge, we propose a novel framework called FABLE (Forecast Alteration By Localized targeted advErsarial attack), which employs a 3D discrete wavelet decomposition to extract the varying components of the geospatio-temporal data. By regulating the magnitude of adversarial perturbations across different components, FABLE can generate adversarial inputs that maintain geospatio-temporal coherence while remaining faithful and closely aligned with the original inputs. Experimental results on multiple real-world datasets demonstrate the effectiveness of our framework over baseline methods across various metrics.