Playing flash videos with Quicktime in Firefox

No.

Expanding on what Kirk said, QuickTime no longer plays Flash content and hasn't in quite a while now. If you need to play Flash content, you'll need to sort out the problems with Flash player. Adobe's forums might be helpful for that.

Regards.

Your answer is better than mine. He's more info:
http://www.adobe.com/shockwave/welcome/

The latest version of Adobe FlashPlayer can be obtained from here:

http://www.adobe.com/shockwave/download/download.cgi?P1ProdVersion=ShockwaveFlash

(You can check here: http://www.adobe.com/products/flash/about/ to see which version you should install for your Mac and OS.

It is suggested that you first uninstall any previous version of Flash Player, using the uninstaller from here (make sure you use the correct one!):

http://kb.adobe.com/selfservice/viewContent.do?externalId=tn_14157

and also that you follow the instructions closely, such as closing ALL applications first before installing. You must also carry out a permission repair after installing anything from Adobe.

You should also ensure that you have downloaded and installed the correct version for your Mac of Security Update 2009-001 (and for Tiger only, 2009-002). (N.B. Security Updates require both a restart and a permission repair.)