Q:

How to increase the number of farmers

I'm already doing the level 3 offline quests and I still only have 1 farmer working for me.  I would have expected an additional farmer by now.
Are there specific quests that I need to do or monsters I need to hunt?

A:

Getting more cats for working the farm seems to be based on just one factor: how many unique quests you have done.  Being in three star village quests, you should be able to get another cat by just completing more quests you haven't done yet.  
The third cat should show up a few quests after the second does.  I believe it's roughly around the 20-24 quest mark.  Three cats is the max, so once you get that last one, the only thing you can do is upgrade your fields.

