I have a fresh installation of elementary OS and I installed "Baldur's Gates II Complete" edition with the original GOG installer.

When I run the game (Fullscreen, Windowed and Setup), it won't start. If I run it in my Terminal, I can see this message:

----
Running Baldur's Gate 2 Complete settings
[drunk][21/12/2018_12:52:26] WINEDEBUG=trace+x11settings
[drunk][21/12/2018_12:52:26] WINEDLLOVERRIDES=mscoree,mshtml=
[drunk][21/12/2018_12:52:26] WINEPREFIX=/home/jonathanlinat/GOG Games/Baldurs Gate 2 Complete/drunk/../prefix/
[drunk][21/12/2018_12:52:26] starting wine
[drunk][21/12/2018_12:52:26] wine run ended
----

How can I solve this problem?

I'd try forcing the game to use the wine package provided by your distribution instead of the one packaged by GOG.

First of all Install wine in Elementary OS by issuing the following command into a terminal: Then search for a file named "drunk.sh" in "/home/jonathanlinat/GOG Games/Baldurs Gate 2 Complete/drunk/" and open it with any text editor.
Replace with Save the file and try to run the game again.