I wouldn’t do it with the garage door opening, but there are lots of ways to do something similar with various physical devices That initiate an event that can be detected by the other system. Usually the easiest one to trigger is a motion sensor on the Honeywell system and then have the SmartThings system turn on a fan or turn on a table lamp Just above the Honeywell motion sensor and that’s usually enough to trigger it. You just have to experiment.
(I’ve mentioned before in the forums how several years ago at a commercial install With multiple incompatible systems we did something similar by starting a remote control toy car from one system that then triggered a motion sensor on a different system. We later replaced this with a fan in a box, but the same idea. 
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You can also have a “robot finger” triggered by one system push a button on the other system. That works, but has a lot more points of vulnerability.
Presumably if anybody has done anything more elegant with software integrations, they will chime in. 
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