The "Zavrsi" underlined link in the bottom right corner of the screen takes the user to a feedback section, clicking that link should end the chat too. Where should I put that link/button? There will be messages in the left side from customer support. Also I cant add that button to the header of chat, because I have a logo and a button to close the chat modal. "Napisite Vasu poruku" is a placeholder for input field.
1- You do not need the "Zavrsi" link to ask users to provide feedback!
You mentioned that you have a Close button in the header! That's all you need. Close the chat and display the feedback questions!
2- If you really want to explicitly ask users to click on a link to provide feedback, check this design by Bank of America's mobile app. They add a little link "Feedback" in a blue color (notice it under Hello!), if clicked it will take the user to a feedback page as the following:
I think the main problem is that everything in your screen looks like a CTA. I think that your current placement is OK, but I'd definitely move away of message bubbles looking like CTA buttons and then add a real CTA for the "Zavrsi" link. Remember that underlined text is a clear affordance for "if I click here I'll go to some page", not for an action.
Furthermore, you mention that you have more actions on the left (it would have been very useful to add them so that we have more context), so if those actions are related to actions after the chat is over, then they should be at the end too.
Otherwise, if they are related to other actions available within the chat, they should probably be within a menu. However, as I said, without seeing those actions is only a speculation.
So this is solution for my problem. This blue box is logo. In this heading on mobile I have X that is closing chat and that X is next to logo. Next to the 'Zavrsi', I have Ref number(it's not on this picture). I think that this solution is the best, because you have X, where you can hide chat, but not lose it's content and you have feedback, where you can finish you conversation and start new one.