Random field geometry has proven relevant results in the context of statistical hypothesis test for solving detection problems in signal and image processing. This paper emphasizes an unsupervised target detection problem in hyperspectral noisy images with very low signal-to-noise ratio (SNR) conditions. The targets have unknown spectral signatures located at unknown bandwidths and positions. To this aim, a spatio-spectral Gaussian random field (SS-GRF) model is proposed to provide a statistical inference about these targets in the full hyperspectral space by means of the geometric features of the noise, notably the expected Euler-characteristic (EC). The performance of the proposed method is demonstrated by the ROC curve analysis on synthetic examples, and confirms its efficiency and capacity to detect hyperspectral targets (astrophysical objects, remote sensing targets). At the end, we discuss the impact of the spectral dimensions on the method.