This paper considers a multivariate quickest detection problem with false data injection (FDI) attacks in internet of things (IoT) systems. We derive a sequential generalized likelihood ratio test (GLRT) for zero-mean Gaussian FDI attacks. Exploiting the fact that covariance matrices are positive, we propose strategies to detect positive semi-definite matrix additions rather than arbitrary changes in the covariance matrix. The distribution of the GLRT is only known asymptotically whereas quickest detectors deal with short sequences, thereby leading to loss of performance. Therefore, we use a finite-sample correction to reduce the false alarm rate. Further, we provide a numerical approach to estimate the threshold sequences, which are analytically intractable to compute. We also compare the average detection delay of the proposed detector for constant and varying threshold sequences. Simulations showed that the proposed detector outperforms the standard sequential GLRT detector.