A conflict conversation is something we've ALL dealt with. It's when the media says or does something which highlights and 'issue' with your perceived identity.
For women it could be abortion, or periods. For Jews it's likely something about Israel. For Hindus it can be the 'Sacred Cow' thing, and for Muslims it can be any hot islamophobic topic in the press at the moment - terrorism, the hijab, al-qaeda, extremism, radicalisation, iraq, afghanistan, guantanamo bay, abu ghraib etc etc.
These things are supposed to push a person's buttons. The questioner is NOT really interested in the topic - if they were they would read about it properly on their own, think about it, and do something about it, and move on. No, they are doing it either to put pressure on you, get a primer (information/soundbite mining) on the topic, or draw you into an argument so that you ideally just meekly accept what they want to say about the issue, or (much more likely) you disagree with them and they can take the moral high ground and say you were 'unreasonable' or 'close minded' or something else which is as dismissive.
And that's the point, it's about being dismissive.
So how do you deal with it??
Simple as ABC: Avoid. Break Balance. Control.
Point out there's plenty of reading material available, and that if they want say a genuinely islamic perspective they are welcome to read the Quran or to investigate the journals in the local library and make up their own mind. They will usually want to avoid the work so they'll say something like 'But, hon, I want YOUR opinion' (they don't) to which the neatest reply is 'My opinion changes the more I read and find out what is actually going on, and I'm sure yours will also.' Then move on. Don't let them start up again.
So now you've somehow found yourself in the discussion, either late entry, or some other way... what do you do? The simplest way is to point out that it IS a conflict conversation, and that the true way to find out about Islam is to spend time with muslims and learn how they live.
Tell the person you will extend an invitation to them to visit the mosque during say ramadan, or to hang out with a few muslims as friends, maybe go to the cinema, take in a ball game, or something normal in every day life. Say that when this person is in the company of more muslims that if it's still on their mind they can raise it. 99% of all conflict conversationalists do not take up this invitation as it would force them to be sincere and show trust in muslims(/jews/hindus/the person they are afraid of really engaging with). If they say they are too busy or the equivalent, say you're also too busy to answer and move on.
NOTE - IT DOES NOT MATTER WHAT THE ISSUE IS, OR WHO THE PERSON IS, DO NOT ARGUE, OR DISCUSS, OR DEBATE, OR RECITE OR ANYTHING.
It will not change their opinion, and may exhaust and frustrate you. Keep that in mind.
Firstly do not ever try to explain the issue, counter intuitive as it sounds this only makes things worse. For both of you.
Listen to the question and no matter what they ask, ask them: What they understand about the issue? Then -crucially- ignore what they tell you and point out their information is incomplete, which you guess is why they asked you.. They will agree. Then ask them why the issue matters to them personally?
They will either say it's of general interest (at which point you say you would otherwise love to discuss it with them but it would take too long and you don't have time/resources to go into the relevant details and point them to a library but - NOT the internet) or make something up (easy to spot, and use the socratic method of exploring the reasons and keeping the focus on their psyche and their motivations until they volunteer to move onto a different conversation - be heartless this is for the best for both of you),
or they come up with a genuine sounding reason, in which case say that any insight you could give would be far less rewarding than reading and thinking about the issue themselves from genuine sources.
Advise them to get a copy of the Quran in English, and also, of course the Hadith, and commentaries. Say they may find it fruitful to read about the history of Islam or spend time in Mosques etc...
And there you go.. your interrogator should either be deflected from their original evil intent, or on their way to genuine enlightenment through their own labours. At the very least they'll think you're not worth the trouble, and hopefully leave you in peace.