They're both correct, but the connotation is different in each. Do you have friends in London? = I'm expecting that you most likely do have friends in London. Do you have any friends in London? = You might not actually have friends in London. I'm not expecting that you will, but you might, and if you do, they're probably not a lot and might be unusual friendships. For example, you can ask someone who has traveled to London quite a lot, "Do you have friends in London?" And you can ask someone who you know knows people from all over the world, "Do you have any friends in London?" Also, think of it this way: if someone smokes you would go to their house and ask "do you have a cigarette?" Whereas if you went to a nonsmoker's house who just recently quit and might still have a pack lying around you'd ask, "do you have any cigarettes?"

The only real difference is that "any" implies "even just one". So as long as you keep that distinction in mind, you can use either one. In a situation where you are in London and you are calling me on the phone to ask for help because you are in trouble there, I would ask, "Do you have any friends in London?" Because even one friend who is there might be able to help you more than I could. I think it would be a little odd in that case for me to ask, "Do you have friends in London," unless I was implying that I didn't want to help you because you probably know a few people there. A different situation. You tell me that you are going to stay at someone's house in London when you visit there. I would ask, "Do you have friends in London?" Because it sounds like maybe you know some people there. It would sound odd for me to ask if you have "any" friends in London in that situation, because I already think you have at least one. What I'm asking is if you have more than one.

Hi! The second one is more grammatically correct because we use "any" with plural nouns, because if it was singular we would say "do you have a friend in london" not "do you have friend in London"

'any' opens the question to all types of answer. Without it the question could be asking for a limited type of friend.