Unfortunately, the devices are unlikely to be compatible. The SmartThings/ADT security panel is a proprietary communication protocol which only works with its own dual logo sensors. Not sensors from other ADT systems.
There are really only two ways to reuse existing sensors. First, if they happen to be Z wave, those are easy: you just reset and then add them to your SmartThings account. If they’re wired, you can use Konnected. But otherwise, it’s like having a system that plays DVDs and finding a stack of old vinyl records. They are just two different technologies.
The following FAQ has more details (this is a clickable link)