The problem of identifying change points in high-dimensional Gaussian graphical models (GGMs) in an online fashion is of interest, due to new applications in biology, economics and social sciences. The offline version of the problem, where all the data are a priori available, has led to a number of methods and associated algorithms involving regularized loss functions. However, for the online version, there is currently only a single work in the literature that develops a sequential testing procedure and also studies its asymptotic false alarm probability and power. The latter test is best suited for the detection of change points driven by global changes in the structure of the precision matrix of the GGM, in the sense that many edges are involved. Nevertheless, in many practical settings the change point is driven by local changes, in the sense that only a small number of edges exhibit changes. To that end, we develop a novel test to address this problem that is based on the $\ell_\infty$ norm of the normalized covariance matrix of an appropriately selected portion of incoming data. The study of the asymptotic distribution of the proposed test statistic under the null (no presence of a change point) and the alternative (presence of a change point) hypotheses requires new technical tools that examine maxima of graph-dependent Gaussian random variables, and that of independent interest. It is further shown that these tools lead to the imposition of mild regularity conditions for key model parameters, instead of more stringent ones required by leveraging previously used tools in related problems in the literature. Numerical work on synthetic data illustrates the good performance of the proposed detection procedure both in terms of computational and statistical efficiency across numerous experimental settings.