Part of your problem is you are using double-quotes when you should be using single-quotes so that, for instance, the $1 isn't interpolated at the time you initialise the $replace variable. Secondly, you need to use the e flag rather than the g flag because you need to execute a bit of code in your replacement part. In fact you need to use a double ee so that execution is done twice, first to interpolate the $replace variable into the "replace" term and second to interpolate again to get what was captured in $1. Thirdly, you have the terms of the substitution round the wrong way; "search" first, "replace" second.

$ perl -le '
> $search    = q{(.*)\d{6}\.txt$};
> $replace   = q{ qq{XXXX_$1} };
> $to_modify = q{AAAA200912.txt};
> print $to_modify;
> $to_modify =~ s/$search/$replace/ee; 
> print $to_modify;'
AAAA200912.txt
XXXX_AAAA
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I hope this is helpful.