Can a site detect if I use AdBlock in Firefox?

Yes, a website can detect that Adblock Plus is installed by referencing a Javascript file known to be blocked by Adblock Plus (because it matches a default rule) and check if this Javascript code has run, by checking a variable for instance.

See an explanation here: How To Detect Adblock Plus

And more details at Wikipedia: Adblock - Detection.

It's entirely possible, it can be done with a bit of javascript. You can see a proof of concept here that shows how to detect multiple different adblock programs, which also uses javascript for detection. It isn't really the fact that Firefox "advertises" your installed add-ons, it's the method that most adblock programs use to function. Most will simply hide elements using the CSS visibility property and filter certain URLs. These URLs can be faked in an attempt to fool the filter, and it is an ongoing battle with regards to blocking detection.