Yes.
You can start git-gui with --trace
flag. On Windows it opens two windows: git-gui and tcl console which also logs executed commands by git-gui [1] [proj. wiki].
You can also open console during running git-gui with CTRL
+ F2
, and turn on logging with: set _trace 1
.
But keep in mind that those commands are "low-level" and shouldn't be used during normal git usage.