Yep, pressure mats are the usual way of doing this. There are number of models available. Some register weights as low as 5 pounds, some go up to the weights of automobiles. The inexpensive ones typically just have an on/off state, with on being triggered at around 25 pounds. Then you attach it to a contact sensor that can communicate with the SmartThings hub when the state changes.
In addition to the project @viguera already shared, there was another topic which started out discussing motion sensors but quickly evolved into a discussion of pressure sensors that would be small enough to fit in a chair. So it might also be of interest.
There’s yet another alternative coming up next year although it’s not yet available for sale. SmartThings has announced a new “sleep sensor” which is a pressure Mat that will go under the mattress. It’s intended for measuring quality of sleep, but it should also give you exactly what you want: an indicator of when someone is in or out of bed. That one will not require any device modification, and should work out of the box with SmartThings.
So if you’re willing to wait for it, and you’re willing to spend more thañ the DIY versions, the sleep sensor might be exactly what you want.
I’m actually quite excited about this one. As someone who is quadriparetic, I spend most of my time in one of three places: my wheelchair, my recliner in the living room, and my bed.
I can think of a lot of different use cases for home automation as I transfer from one to the other. 
On the other hand, if you want to keep your budget as low as possible, options mentioned in the Smart Chair topic are probably the least expensive.