As dialogue systems become increasingly important across various domains, a key challenge in persona-based dialogue is generating engaging and context-specific interactions while ensuring the model acts with a coherent personality. However, existing persona-based dialogue datasets lack explicit relations between persona sentences and responses, which makes it difficult for models to effectively capture persona information. To address these issues, we propose MoCoRP (Modeling Consistent Relations between Persona and Response), a framework that incorporates explicit relations into language models. MoCoRP leverages an NLI expert to explicitly extract the NLI relations between persona sentences and responses, enabling the model to effectively incorporate appropriate persona information from the context into its responses. We applied this framework to pre-trained models like BART and further extended it to modern large language models (LLMs) through alignment tuning. Experimental results on the public datasets ConvAI2 and MPChat demonstrate that MoCoRP outperforms existing baselines, achieving superior persona consistency and engaging, context-aware dialogue generation. Furthermore, our model not only excels in quantitative metrics but also shows significant improvements in qualitative aspects. These results highlight the effectiveness of explicitly modeling persona-response relations in persona-based dialogue. The source codes of MoCoRP are available at https://github.com/DMCB-GIST/MoCoRP.