If you have a camera close to your garage, you can plug a bridge into that camera and the sensors should work. The range on the sensors have a limit especially if there are several walls the signal has to go through. Some of mine are about 24 feet away from my camera and they still work fine.
Your guess is correct. The sensors and motion detection devices are not wifi devices and only communicate with a bridge that is in a camera and they have a range of 25 30 feet so you will have to actually get a bridgeon your camera in the garage and you should solve your problem
I’m running one motion sensor over 100’ away from the bridge, through 1/2" plywood and the front wall of the house with a large glass window. So long distances are possible. As others have mentioned, how much and what type of materials it is transmitting through make a big difference. Stone fireplace, wall and bookcase sounds like a lot. And that doesn’t count the garage walls, right? Speaking of which, does the garage perhaps have metal siding? That would really reduce the range.