I hope that another war between North Korea and South Korea won't break out! (Normally it would be "a war", but since this is war has already happened in the past and might happen again, it'd would be "another war".) 

Lately, the relationship between North Korea and South Korea has been increasingly intense. I really hope there won't be any wars breaking out down the road!

(I wanted to correct what Nancy wrote. What you wrote was right: it is "relationship...has been", not "relationship...have been", because it's only one singular relationship. "Lately" can have a comma after it or not, either way is correct. I put "really hope" instead of "do hope" or just "hope" because it seems like you want to emphasize that this is your sincere feelings, and it sounds most natural this way.)

(Btw, I find it interesting the difference in perspective between us or perhaps, our countries when it comes to the meaning of "Korea". When Americans say "Korea", we think "South Korea". Only when we mean "North Korea" do we say that specific direction of the country. But you seem to use "Korea" to mean "North Korea", and speak of South Korea with the specific direction. That's interesting!)

I hope a war between North Korea and South Korea doesn't break out!

Lately, the relationship between North Korea and South Korea have been increasing in intensity.
I do hope a war doesn't break out down the road!

"do hope" is very UK (sound sillly outside of the UK. <
.