Yep - I was using an emulator which didn't have Google Play Services installed. So, after getting some advice, I updated my Android SDK and switched to a higher-version API emulator (API 26) which does have Google Play Services installed. However, B4A and the higher-version emulator won't connect. When looking at the ADB entry, the emulator shows up as "unauthorized".
Just as a further note, I've discovered that this only occurs with API 26. With a lower API, it works (the "unauthorized" is replaced with "device").