No, you shouldn't need to do anything else. As long as that cable is working properly, it should connect through it and "just work". To check on it, use the Google Home app and look at the details for that remote Google WiFi unit in the garage. Under "Wi-Fi Information", it should say "wired" next to "Connection type". I would probably also run a mesh test after making the connection and verifying it says "wired" for that secondary/point and make sure the test results say "Great connection" for it. If it is wired, but the mesh test says "weak", the cable may only be managing to carry 100Mbps instead of the full 1Gbps. If that happens, test that cable to be sure all four twisted pairs are working properly. I would also think about putting an inexpensive unmanaged Ethernet switch on each end of that cable run. That will provide some electrical isolation (better to replace a couple of $25 Ethernet switches than have to replace a couple of router/point units after a lightning strike), and having those switches will make more Ethernet ports available for other equipment in both locations.
