In Firefox, the default encoding is in Options|Content, in the dialog box shown by Advanced... under Fonts & Colors.
However, the server can also specify, in the headers, the character set it is using. From a quick web search, user agents (that is, browsers) can choose to return POST content using the same value. It is also possible to explicitly specify the character set to be used in a FORM (i.e. the type-in boxes for posting messages).
Inspecting the content of this page, the Quick Reply box does just that; see the following. Note the accept-charset="UTF-8". That means that, at least for that box, the browser must send UTF-8 to the server.
<form action="http://www.apollohoax.net/forum/index.php?action=quickmod2;topic=336.15" method="post" accept-charset="UTF-8" name="quickModForm" id="quickModForm" style="margin: 0;" onsubmit="return oQuickModify.bInEditMode ? oQuickModify.modifySave('ff2e0a69ff2041e702688f31cd533f0f', 'adc929d5') : false">
For reference, here are the headers supplied by this server; note that the character set is UTF-8.
HTTP/1.1 200 OK
Date: Mon, 28 Jan 2013 18:39:48 GMT
Server: Apache/2.2.22 (Unix) mod_ssl/2.2.22 OpenSSL/0.9.8e-fips-rhel5 mod_auth_passthrough/2.1 mod_bwlimited/1.4 FrontPage/5.0.2.2635 mod_fcgid/2.3.5
X-Powered-By: PHP/5.3.15
Pragma: no-cache
Cache-Control: private
Expires: Mon, 26 Jul 1997 05:00:00 GMT
Set-Cookie: PHPSESSID=a1caf8341533381e6b1f79998bce9669; path=/
Last-Modified: Mon, 28 Jan 2013 18:39:48 GMT
Connection: close
Content-Type: text/html; charset=UTF-8
ETA: Bob B., what browser are you using? (product, e.g. IE, and version, e.g. 9).