If you are using anti-virus software, it is quite possible that it is the culprit. Anti-virus software can block applications that it thinks are "suspicious" even if the applications are actually legitimate. Can you tell us which anti-virus software you are using?
Also, if you type this into Status Window editbox in mIRC:
/server irc.quakenet.org 6667
And then press the enter key, mIRC should connect to the QuakeNet IRC network. If it does not connect, can you tell us what error message you are seeing in the Status Window?