The MySQL Query Browser is a really powerful tool once you get to know it. The information you're looking for is in the zones table, but you probably don't want all of the data that it provides if you're just looking for zone IDs. Here's a quick tip:
In the pane on the right side, you have a list of all the tables in the database. Double clicking the name of a table and clicking execute (or hitting CTRL Enter) runs the query, showing you all of the columns and rows for the table. But say you only need a few...?
Left click the arrow to the left of the zones table. This expands it, so that you can see all of the columns. Double click the ones you'll need. For what you're asking, I'd double click short_name, long_name, and zoneidnumber. The query at the top should look like this:
Code:
SELECT z.`short_name`, z.`long_name`, z.`zoneidnumber` FROM zone z;
Execute it, and you should get three columns showing you exactly what you want.