Can a property manager garnish a tenant's wages if they owe back rent?

Yes they can. They can put it in the hands of a Debt Collector (small debts) and even take you to court. It would be to your best interest to quit running from your problem and try to work out something with the Property Manager. By running, they'll just catch up to you eventually. Face your consequences and hopefully you have a good reason for being behind in your rent. Always make good on your debts even if you have to bargain to pay that debt off! When someone wins a lawsuit, one option is to seek payment through through the method you described.