Not a Perl solution, but you can simply unquote the var:

ACTION_PATTERN='perl$'
ROWS=$(perl -n -e '$count++ if /'$ACTION_PATTERN'/; END { print $count
+ }' ~/.zsh_history)
echo "ROWS:${ROWS}"
[download]

Caveat: only tested in bash. I'd be surprised if it didn't work in zsh too, though.