In recent years, the success of large language models (LLMs) has inspired growing interest in exploring their potential applications in wireless communications, especially for channel prediction tasks. However, directly applying LLMs to channel prediction faces a domain mismatch issue stemming from their text-based pre-training. To mitigate this, the ``adapter + LLM" paradigm has emerged, where an adapter is designed to bridge the domain gap between the channel state information (CSI) data and LLMs. While showing initial success, existing adapters may not fully exploit the potential of this paradigm. To address this limitation, this work provides a key insight that learning representations from the spectral components of CSI features can more effectively help bridge the domain gap. Accordingly, we propose a spectral-attentive framework, named SCA-LLM, for channel prediction in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. Specifically, its novel adapter can capture finer spectral details and better adapt the LLM for channel prediction than previous methods. Extensive simulations show that SCA-LLM achieves state-of-the-art prediction performance and strong generalization, yielding up to $-2.4~\text{dB}$ normalized mean squared error (NMSE) advantage over the previous LLM based method. Ablation studies further confirm the superiority of SCA-LLM in mitigating domain mismatch.