Energy harvesting (EH) is an emerging technology for enabling sustainable wireless communication in sixth-generation (6G) networks, particularly for energy-constrained network nodes that are difficult to maintain or recharge. In this paper, we propose and analyze an adaptive energy-harvesting decode-and-forward (DF) relaying framework over generalized $\alpha$-$\mu$ fading channels. The proposed framework consists of a multi-antenna source, an energy-constrained multi-antenna relay, and a multi-antenna destination, with direct source-to-destination communication assumed infeasible due to severe channel impairments. To sustain relay operation, the relay harvests energy from received radio-frequency (RF) signals using time-switching (TSR) and power-splitting (PSR) protocols, then forwards the decoded information in half-duplex mode. The generalized $\alpha$-$\mu$ distribution provides a unified statistical characterization of wireless propagation across diverse 6G scenarios and encompasses several well-known fading models as special cases, such as Rayleigh, Nakagami-m, Weibull, one-sided Gaussian, and exponential fading. The study considers multiple transmission strategies, including optimal rate adaptation (ORA), optimal power and rate adaptation (OPRA), channel inversion with fixed rate (CIFR), and truncated channel inversion with fixed rate (TIFR). Exact-form expressions for the outage probability, Bit error rate (BER) using the moment-generating-function approach, and channel capacity are derived for the considered adaptive transmission strategies. The diversity order is also characterized to evaluate system performance under different numbers of antenna elements, transmit power levels, energy-harvesting ratios, and fading severities. Numerical results from Monte Carlo simulations demonstrate the performance of the proposed framework under different system configurations, including Amazon Kubernetes Service (AKS). The analytical results closely match the simulation results, validating the derived expressions. Overall, the proposed EH-enabled Decode-and-Forward (DF) relaying framework provides a flexible approach to improving reliability, reducing outage probability and BER, and enhancing channel capacity in energy-constrained 6G wireless networks.