If you use grep -c, you'll get a count for each file, and then you'd have to add it. I guess doing a grep first and then piping it to wc is the easiest way. You can redirect stderr to /dev/null to skip error messages:
Code:
grep zamo */*/* 2>/dev/null | wc -l
(I just tried this, and wc cleverly counts only the lines from stdout. So your count will remain the same whether or not you suppress the error messages).