first, a couple command-line solutions:

#bash:
for f in `find /tmp/files/`; do head -20 $f | grep -q -i dave && echo 
+$f >> list.txt ; done

perl -lne 'do { print $ARGV; last } if /DAVE/i; last if $. == 20'  `fi
+nd /tmp/files`  >> list.txt
[download]

Now, addressing your code, couple starting points: note that $. (see perlvar) can be used instead of $count ; instead of string matching 'Dave' (which is why the case has to be exact), use a regular expression of /dave/i (or maybe /\bdave\b/i) -- see perlre for tons of regexp info.
Also, why use File::Find if you're given a single filename to start with?