I'm not a PHP or MySQL developer, but it sounds like you ought to just be changing DATE
in your original code with YEARWEEK
WHERE YEARWEEK(CONVERT_TZ(from_unixtime(comment_date), '$timezone1', '$timezone2'))
= YEARWEEK(CURRENT_DATE, 1)";
If that's not working, please give more information about the way in which it's not working.
Also note that you're not really providing time zones - you're providing offsets. That means your code will give inconsistent results based on daylight saving time. This may or may not be a problem, depending on your requirements.