I'm not sure how to change the Content-Type header with HTTP::Request::Common (for plain WWW::Mechanize), but it should give you at least a HTTP::Request whose Content-Type you can change later on:

use strict;
use HTTP::Request::Common;
my $request= POST 'http://book.flypeach.com/WebService/B2cService.asmx
+/SearchLowFareSingleMonth',
         Content_Type => 'application/json',
         Content => '{"strFromAirport":"KIX","strToAirport":"HKG","dep
+artMonth":"20150606","returnMonth":"20150606","iOneWay":"true","iAdul
+t":2,"iChild":0,"iInfant":0,"BoardingClass":"","CurrencyCode":"JPY","
+strPromoCode":"","SearchType":"FARE","iOther":0,"otherType":"","stIpA
+ddress":"","strCurrentDate":"20150406"}',
       ;
print $request->as_string;
__END__
POST http://book.flypeach.com/WebService/B2cService.asmx/SearchLowFare
+SingleMont
h
Content-Length: 295
Content-Type: application/json

{"strFromAirport":"KIX","strToAirport":"HKG","departMonth":"20150606",
+"returnMonth":"20150606","iOneWay":"true","iAdult":2,"iChild":0,"iInf
+ant":0,"BoardingClass":"","CurrencyCode":"JPY","strPromoCode":"","Sea
+rchType":"FARE","iOther":0,"otherType":"","stpAddress":"","strCurrent
+Date":"20150406"}
[download]

If you want to keep using WWW::Mechanize::Firefox, note that its support to add custom headers is somewhat limited, but the same approach could still work. But if you already use WWW::Mechanize::Firefox, why not just keep automating the complete website and programmatically click your way to the data you want?