First we must determine whether your motion sensors are wired or wireless.
The wired PIR motion sensors used in security systems since roughly forever are pretty simple devices. You feed them 12 volts dc to power the electronics; they contain a relay which opens or closes to indicate motion. Some will have a SPDT relay so that the installer can choose whether to use the normally open or normally closed contact. "Normally" means the inactive or resting state of the device, ie no motion detected.
If your ADT system is wired there'll be a control panel buried in a closet or utility room somewhere in the house. Open it up and you'd most likely find a 4-conductor cable for each motion sensor. If the motion sensors are grouped into zones then it's possible you'd have one cable for each zone; each sensor might not have its own "home run" cable back to the panel.
All these wired sensors could be fed into some kind of multi-input interface. That might be built around an Arduino, a Raspberry Pi, or some other experimenter/hobbyist platform.
If your motion sensors are wireless then things aren't necessarily harder, just different. First you'll have to figure out what RF frequency and protocol they use. At very least they should be labelled with an FCC ID number which reveals the manufacturer and model of the device, and from there the rest can be discovered. They might also simply be labelled such that a person familiar with the industry would recognize them. GE/Interlogix is one particular make that's quite prolific. Each sensor has a unique serial number which it periodically transmits along with its battery state, contact state, and other info. Receivers are available that can listen for these sensor transmissions and send it over a serial bus like RS-232.
Pulling it together
Identifying what sort of sensors you're working with is the easy part. From here you're probably going to be looking for some kind of open source home automation platform such as openHAB, Home Assistant, etc. These may provide a software framework for receiving the security sensor inputs and putting them into a web or mobile app front end you can use. This all goes beyond the scope of the DIY stack exchange, but hopefully I've given enough ideas and key words to help your research get started.