Try this:

#!/usr/bin/perl

use strict;
use warnings;
use diagnostics;

my $file = $ARGV[0];
open(my $fh, '<', $file)
  or die "$file: $!";

my $query;
while (<$fh>) {
    if (/^Query=\s(\S+)/) {
        $query = $1;
    }
    if (defined($query) && /^>(\S+)/) {
        print "$query\t$1\n";
        undef $query;
    }
}
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