Do you need ESD protection? That really depends on what the H-bridge is connected to. There are a few sources of ESD that you might worry about.
The most likely one is people, a few thousand volts (8kV for the human body model) is what you need to prepare for if whatever is going to be connected to the h-bridge is exposed to people. It really depends on if the device you have is enclosed. If it is more like a faraday cage with some kind of conductive material around it, you won't have much of a problem. I've drawn a circuit to give some ways to think about ESD. Air has picoFarads of capacitance, when something is charged up like a finger, it starts charging up the air around it. if you get to close the capacitor breaks down and shorts. The thing that matters is what is on the inside of your box. Is there a motor exposed to the outside world that could be a low impedance pathway for the ESD to get into your circuit like R4? Are there screws like R1 connecting to ground for the ESD to make a return path through your product? If your case is insulative all the way around then there won't be much opportunity for the ESD to go through it. But if you had a motor connected to the h-bridge and some screws running through the case then there might be a better pathway and you would have to worry about it. The best case scenario is a metal case with all of your electronics on the inside. If a person is holding it in there hand, then everything will be the same potential and it wont matter. The worst case scenario is if was sitting on a metal desk and the lowest impedance pathway is through the electronics. So you may need ESD protection, but it depends on the pathway. Also the most sensitive part of the mosfet to ESD is the gate. And make sure the pathway of the ESD through the protection and out to the chassis ground.
simulate this circuit – Schematic created using CircuitLab
The second one is triboelectric charging from rubbing one surface on another. Cables can do this, if you have a cable on your device you could run into both people generating friction on the cable and charging it up and shocking it. This would definitely require protection.
You can also do ESD testing they have calibrated guns that you use to simulate a human shock. If you can't afford one of these a pizo electric shocker can work, if you are up for some potentially destructive testing. If it was a cost issue, I would test it and see if you have a problem. Caps can also soak up ESD to some extent, if the motor had a high impedance and a low capacitance, there wouldn't be much current running through it and some 1nF caps might do the job, but I would test it.