Backchannels and fillers are important linguistic expressions in dialogue, but often treated as 'noise' to be bypassed in modern transformer-based language models (LMs). Here, we study how they are represented in LMs using three fine-tuning strategies on three dialogue corpora in English and Japanese, in which backchannels and fillers are both preserved and annotated. This allows us to investigate how fine-tuning can help LMs learn these representations. We first apply clustering analysis to the learnt representation of backchannels and fillers, and find increased silhouette scores in representations from fine-tuned models, which suggests that fine-tuning enables LMs to distinguish the nuanced semantic variation in different backchannel and filler use. We also employ natural language generation metrics and qualitative analyses to verify that utterances produced by fine-tuned LMs resemble those produced by humans more closely. Our findings suggest the potential for transforming general LMs into conversational LMs that can produce human-like language more adequately.