You might be able to do this with Google Drive. But you'll need to make sure you only have one application open at a time, otherwise you could corrupt the database. Here's what I would recommend:
1. Create a google.com account
2. Download Backup and Sync on both Macs
3. Make a backup copy of Astroplanner and store in somewhere else on one of the Macs
4. Copy the astroplanner database files to Google Drive
It should sync between both machines. A big note. Astroplanner is 32-bit software and will no work on macOS Catalina (comes out this fall). So unless the developer releases a 64-bit version of the software do not upgrade your Macs or you won't be able to run the software. I used Astroplanner for about 10 years and loved it. But it's not really maintained anymore so I've moved on to SkySafari and LiveSky. It does most of what Astroplanner does, but definitely not as low-cost (buy once vs. subscription).