Sorry this turned out to be such a serious talk for our first meeting.
I didn't mean for this to be such a heavy conversation, especially since this is the first time we met.
ーSorry this turned out to be such a serious talk for our first meeting.
「初対面なのにこんな真面目な話になっちゃってすみません」
to turn out to be で「（結果）〜となる」
a serious talk で「真面目な話」
one's first meeting で「初対面」を表現できます。
ーI didn't mean for this to be such a heavy conversation, especially since this is the first time we met.
「こんな重い会話にするつもりはなかったんです。特に今日は初対面だったので」
a heavy conversation で「重い会話」
first time we met を使って「初対面」を表現することもできます。
ご参考まで！
カナダ 
