You can roll your own quoting for POSIX-like shells fairly simply - no complicated regexes needed (just straightforward string substitution using literals):
$ echo "I'm \$HOME. 3\" of rain." | perl -lne "s/'/'\\\''/g; print q{'} . \$_ . q{'}"
'I'\''m $HOME. 3" of rain.'
The approach is modeled after AppleScript's quoted form of command:
The input string is broken into substrings by ', each substring is itself '-enclosed, with the original ' chars. spliced between the substrings as \' (an individually quoted ').
When passed to the shell, the shell rebuilds these parts into a single, literal string.
This multi-part string-concatenation approach is necessary, because POSIX-like shells categorically do not allow embedding ' itself inside single-quoted strings (there's not even an escape sequence).
Alternatively, you can install a CPAN module such as ShellQuote.
Optional background information
While it would be handy for Perl itself to support such a quoting mechanism for piecing together shell commands stored in a single string to pass to qx// (`...`), such a mechanism would have to operate platform-specifically.
Notably, quoting rules for Windows are very different from rules for Unix platforms, and except for simple cases shell commands as a whole will be incompatible too.
From inside Perl, you may be able to bypass the need for quoting altogether, by using the list forms of system() and open(), which allow you to pass the command arguments individually, as-is, but note that this is only an option if your command doesn't use any shell features; for a "shell-less" qx// (`...`) alternative, see this answer of mine, which also covers shell-quoting on Windows.