~してもいい is commonly used to give or request permission, e.g. 「入ってもいいですよ。」"You may come in." (giving permission) / 「電話を借りてもいいですか?」"May I use your phone?" (requesting permission). But in your example:
紹介してもいいんですけれどもね。
the してもいい means "I am willing to do~~" "I am ready to do~~", rather than "I may~~ / I am allowed to~~".
"I am willing to introduce (someone/something)(to someone), but..."
The ~けれどもね at the end, literally "Although..." or "... but", adds a nuance of uncertainty, hesitation or maybe reluctance, like "...but only if~~" or "...but I'm not sure~~", depending on the context.