If you have this on a dedicated file-system, or you have a steady number of files overhead, you may be able to get a rough enough count of the number of files by looking at the number of inodes in the file-system via "df -i":
root@dhcp18:~# df -i
Filesystem Inodes IUsed IFree IUse% Mounted on
/dev/sda1 60489728 75885 60413843 1% /
On my test box above I have 75,885 inodes allocated. However, these inodes are not just files, they are also directories. For example:
root@dhcp18:~# mkdir /tmp/foo
root@dhcp18:~# df -i /tmp
Filesystem Inodes IUsed IFree IUse% Mounted on
/dev/sda1 60489728 75886 60413842 1% /
root@dhcp18:~# touch /tmp/bar
root@dhcp18:~# df -i /tmp
Filesystem Inodes IUsed IFree IUse% Mounted on
/dev/sda1 60489728 75887 60413841 1% /
NOTE: Not all file-systems maintain inode counts the same way. ext2/3/4 will all work, however btrfs always reports 0.
If you have to differentiate files from directories, you're going to have to walk the file-system and "stat" each one to see if it's a file, directory, sym-link, etc... The biggest issue here is not the piping of all the text to "wc", but seeking around among all the inodes and directory entries to put that data together.
Other than the inode table as shown by "df -i", there really is no database of how many files there are under a given directory. However, if this information is important to you, you could create and maintain such a database by having your programs increment a number when they create a file in this directory and decrement it when deleted. If you don't control the programs that create them, this isn't an option.