Another variant, using Text::Balanced, which is also always useful for parsing stuff with quoted, parenthesized, etc. parts:

#!/usr/bin/perl -l
use strict;
use warnings;
use Text::Balanced qw(extract_delimited);

my $text = 'SELECT "SELECT * FROM table1 WHERE ..." SQL FROM table2 WH
+ERE ...';

my (undef, $remainder) = extract_delimited($text, '"', '[^"]*');

my ($extracted) = $remainder =~ /(\bFROM\b.*?\bWHERE\b)/;

print $extracted;   # FROM table2 WHERE
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