・chat=ちょっとした会話、雑談
(例)
I had a nice chat with her today. She was very friendly.
Would you like a chat about it sometime soon?
・conversation=会話
(例)
I had a conversation about my grades with my parents.
・lively=活発な、明るい、弾む
(例)
It was a very lively atmosphere!
お役に立てれば幸いです。
「深む」が"deepened"と訳し、「膨らむ」を"expanded"と訳しました。
または、"go into depth"「(会話の内容などが)深いところまで行く」という言い方もできます。"expanded"の代わりに、"developed"「発達した」、またはカジュアルな言い方だと、"take off"という言い方も使えます。
Sentence in context:
"We became very good friends after one particular conversation we had in which it explored the subject in depth and our conversation took off."