I don't know the sentence in Russian, but "finish up" is not quite right when talking about a conversation.
"Finish up" is more like "complete" or "do the last bit"

Here is how it's used:

-Have you finished up in there?
-Yes it's completely finished now. Let's start cleaning out the next room.
-He's just finishing up in the garage. He'll come in for dinner in five minutes.
(the person has been doing something in the garage and is just about to finish and leave).
This is a minor detail in native level English. It does not matter if you do not know this, there is practically no difference between saying "finish up "and "finish". If you are not sure how to use "finish up" then there is no particular reason to use it.