When an application crash without output an error, is there a log that i can check?

Depends on the application. Different applications have different logging systems; there's no one central log that contains all the output from all the programs that run on your system.

That being said, a lot of programs do put their log files in the directory /var/log. The file /var/log/syslog (or maybe /var/log/messages), in particular, contains output from the "system logger", which is a service made available by the system that programs can use (if they choose to) for logging. But not all programs use it. Mostly, you'll find messages from low-level system services in that file, not the graphical applications you probably use normally.

You may want to read more about the locations of standard log files.

crash files go into /var/log/crashes/ for use with apport to report bugs. You can extract a core dump with apport-unpack, put that core dump through gdb, and find out what's causing the program to crash.

This is all assuming you're a programmer. If you're not...well, you can't fix the crash anyway!

For conky it could also be that there are entries in $HOME/.xsession-errors.