Translating legacy Fortran code into C++ is a crucial step in modernizing high-performance computing (HPC) applications. However, the scarcity of high-quality, parallel Fortran-to-C++ datasets and the limited domain-specific expertise in large language models (LLMs) present significant challenges for automated translation. In this paper, we introduce Fortran2CPP, a multi-turn dialogue dataset generated by a novel LLM agent-based approach that integrates a dual-LLM Questioner-Solver module to enhance translation accuracy. Our dataset comprises 11.7k dialogues capturing iterative feedback-decision workflows including code translation, compilation, execution, unit testing, and error-fixing. Using this dataset, we fine-tune several open-weight LLMs and achieve up to a 3.31x improvement in CodeBLEU scores and a 92\% increase in compilation success rate, demonstrating enhanced syntactic accuracy and functional reliability. Our findings highlight the value of dialogue-based LLM training for complex code translation tasks. The dataset and model have been open-sourced and are available on our public GitHub repository\footnote{\url{https://github.com/HPC-Fortran2CPP/Fortran2Cpp}}.