This paper studies the device activity detection problem in a massive multiple-input multiple-output (MIMO) system for near-field communications (NFC). In this system, active devices transmit their signature sequences to the base station (BS), which detects the active devices based on the received signal. In this paper, we model the near-field channels as correlated Rician fading channels and formulate the device activity detection problem as a maximum likelihood estimation (MLE) problem. Compared to the traditional uncorrelated channel model, the correlation of channels complicates both algorithm design and theoretical analysis of the MLE problem. On the algorithmic side, we present the classical exact coordinate descent (CD) algorithm for solving the MLE problem, which suffers from numerical instability when applied to correlated channels. We propose a computationally efficient inexact CD algorithm by approximating the objective function, which approximately solves the one-dimensional subproblem and improves both computational efficiency and numerical stability. Additionally, we analyze the detection performance of the MLE problem under correlated channels by comparing it with the case of uncorrelated channels. The analysis shows that when the overall number of devices $N$ is large or the signature sequence length $L$ is small, the detection performance of MLE under correlated channels tends to be better than that under uncorrelated channels. Conversely, when $N$ is small or $L$ is large, MLE performs better under uncorrelated channels than under correlated ones. Finally, we study the MLE model in the joint device activity and data detection context. Simulation results demonstrate the computational performance of the presented algorithms and verify the correctness of the analysis.