First, you will need to create a Sound folder in your WoW folder. Here is the folder structure for mine:
\World of Warcraft\_retail_\Sound
There are subfolders under the Sound folder that place the below mentioned blank files into the correct path. Here is the path for Mimiron’s files:
\World of Warcraft\_retail_\Sound\Creature\MimironHeadMount
Now, you will need to create blank sound files for these specific sounds. To do that, just create 4 blank text files and name them the following:
MimironHeadMount_JumpEnd.ogg
MimironHeadMount_JumpStart.ogg
MimironHeadMount_Run.ogg
MimironHeadMount_Walk.ogg
So, just take those 4 files you just created and place them into the full folder path shown above and that should silence the Mimiron mount.