undo-tree-save-history calls write-region. In this example, we suppress the Wrote ... message entirely using an :around advice. Here are the steps used to verify the answer works as advertised:
STEP 1: Launch a recent public release of Emacs 26.3 -- without any user-configuration, aka emacs -Q
STEP 2: Download the library undo-tree, which was last ...
A few years ago, I created a fork of undo-tree that was compatible with Emacs 25 (but not later/newer versions of Emacs) and I spent a considerable amount of time debugging in relation thereto. As part of that work, I inspected and debugged countless undo-tree saved history files -- that prior work helped me to identify the initial cursory debugging message ...
Since answering this question, evil now supports pluggable undo systems.
So if you want to use undo-fu instead of undo-tree you can do this as follows.
(setq evil-undo-system 'undo-fu))