First off:
$query = User::where('id', 1);
This starts a Query Builder instance. To get the object you're looking for use:
$query = User::where('id', 1)->firstOrFail();
Now you need the username to update to the id:
User::where('id', 1)->update(['ssowat_id', => $query->username]);
Second, never use Models within migrations. Rather use raw queries, eg:
\DB::table('users')->findOrFail(1);
Not sure whether DB is accessible here though ?
Now that we covered that, I'm not understanding what you're trying to achieve. I guess there will be a simpler solution than what you try to do here.