No, you don’t need to stop paying your mortgage to qualify for a loan modification. You only need to show you are in danger of falling behind in your payments.
In fact, purposely falling behind on your mortgage payments can become a big problem for you in the future. For one thing, it will significantly lower your credit score. Plus, past due amounts will continue to add up and eventually you will have to pay those costs, along with interest and late fees. If there’s any way you can make your mortgage payments, you should continue to do so.