This study is focused on understanding and quantifying the change in phoneme and prosody information encoded in the Self-Supervised Learning (SSL) model, brought by an accent identification (AID) fine-tuning task. This problem is addressed based on model probing. Specifically, we conduct a systematic layer-wise analysis of the representations of the Transformer layers on a phoneme correlation task, and a novel word-level prosody prediction task. We compare the probing performance of the pre-trained and fine-tuned SSL models. Results show that the AID fine-tuning task steers the top 2 layers to learn richer phoneme and prosody representation. These changes share some similarities with the effects of fine-tuning with an Automatic Speech Recognition task. In addition, we observe strong accent-specific phoneme representations in layer 9. To sum up, this study provides insights into the understanding of SSL features and their interactions with fine-tuning tasks.