There is no option not to print the file name, you'll need to use a pipe as phil.o showed. Some other options:
#use cat
cat my_file | wc -c
#use cut
wc -c my_file | cut -f1 -d ' '
#you could also use stat, if you don't want to use a pipe
stat -c "%s" my_file
If your file is UTF-8, and you need to get a count of the glyphs, then you probably want to stick with
wc -m
I'm sure there's a number of other ways to get that answer too.