If the existing client is already a friend of yours to the point where you would feel comfortable sharing your work with them, that's one issue. If the existing client is not someone you already know, that's something else.
For the first one, I'm not sure how to handle that. I have friends who have beta-read for me who have agents I'd like to query, but I've honestly never asked for a referral so I don't know how you do that. I imagine you handle it with delicacy and tact, and ask if they think it's something their agent might be interested in. Or wait until they bring it up. But again, I don't know.
For the second one, that's actually a big no-no. Do not approach someone you don't know asking about their agent. They don't know you and have no basis on which to judge your work. And it's not appropriate to ask someone you don't know to read your work if they haven't come into the situation looking for that kind of thing (like at a workshop or on here, for example). It's pushy and unprofessional, in my opinion.
Your best bet, as others have said, is to try to meet these agents at a conference or workshop. Otherwise, there are many, many good agents who are open to queries. Don't get fixated on one dream agent because that person might not actually be a good fit.