You can use positive zero-width assertions:

my $t = q["string (12" or 1 foot or 1') into 6" MySQL varchar"];

$t =~ s/(?<=.)"(?=.)/""/g;

print "$t\n";
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This means literally 'A double-quote that is preceded by a any char and followed by any char'. See perldoc perlre for the details.

•Update: corrected terminology
•Update: if the string can contain newlines, don't forget to add the /s modifier to the substitution