"Your friend is handsome, can you introduce him to me?"
I think they're both correct. The difference being 介绍我 implies giving an introduction "of" you (instead of "to" you), like if someone is introducing you as a speaker at a conference. But if you use 给我介绍, you're asking the person to make an introduction of the friend, to you as the audience.
会 is about ability and similar to "is able to". (Ex: "I can speak Mandarin." or "I can't swim.") 能 is about possibility and is similar to "is possible to". (Ex: "I'm feeling better, I can go out tonight." or "He can't make it back in time, he's too far.") 可以 is about permission and is similar to "is allowed to". (Ex: "I can park here." or "We can't stand in the VIP line.")
Its the same reason that you can't really say stuff like “我高” to mean "I'm tall", I think. You need to say "我很高“. The reason I see is always something to do with natural phrasing and spacing. It's true that "高” can mean "..to be tall", but a chinese person will want something between 我 and 高, apparently, and the most common thing is 很. So while people would understand you and probably not care if you said “你的朋友帅” to mean "your friend is handsome", a native speaker would never say it this way, probably.
My answer marked incorrect: 你的朋友很帅，可以介绍他吗？ The accepted answer: 你的朋友很帅，可以给我介绍吗？
I'm not seeing anything in these comments, or elsewhere online, as to why my translation would be incorrect. If anyone has any ideas, I'd be interested in hearing them.
This may be bias talking, but I think my translation is not only correct but superior since it specifies the gender of the friend while the accepted answer doesn't.
I'd translate the accepted answer as: Your friend is very handsome, can you introduce me?