No, that would be incorrect to say "the 20's..."
There isn't (to my knowledge) a word or phrase that expresses the third week of the month other than simply saying "the third week of the month".
In business, you will often hear people referring to a week of the year by its number (there are 52 ). Today, we begin week 2 of 2012.
Week Numbers for 2012
Interested in Language