Run Quicken Mac on Two Computers?

It's definitely possible, but it requires a little work on your part. Quicken is built to have your data file on your local computer; while it can sync some information for use with the mobile app or web interface, it was not designed to fully sync your entire data file between multiple computers. To do that, you need to move your Quicken data file back and for the between the two computers. You can use a cloud storage service to make this pretty easy -- as long as you do it correctly. 

Some basic do's and don'ts about moving a Quicken Mac data file around...

Do not store your Quicken data file on any cloud service, such as Dropbox or iCloud. It will likely lead to problems. Storing backup files on cloud storage is fine, but your live data file needs to be resident on your local computer. 

Do move only a compressed copy of a data file. A compressed file is either (a) one you create by selecting the file in the Finder and doing File > Compress, creating a .zip file, or (b) a backup file created by Quicken, which ends in .quickenbackup. Why? A Quicken data file is actually not a single file; it's a Mac "package" file, which is a wrapper around a collection of files and folders to make it appear to users like a single file. (Control-click on your data file and select "Show Package Contents" if you want to peek inside the wrapper.) Every Mac user account has a unique User ID number, and when you move files and folders around, permissions can be changed -- the result of which can be getting locked out of your data. Moving a compressed file and opening it on a different Mac won't result in permission problems. (Moving a compressed file can be either via a cloud service, a local network, Airdrop, or a flash drive.) 

This may sound like a pain, but it needn't be. After each time you use Quicken, move your backup or compressed file to a location -- on cloud storage or a physical flash drive --  you'll start from the next time you use Quicken on either computer. Use the data file, quit Quicken, and again save the compressed file back to the same location. As long as you always start from the same location and replace a file to the same location -- which takes just a few extra seconds -- you will always be assured of working on the most current file and not having permission problems. 

If you use Dropbox, one relatively easy workflow is to set your Quicken backup location to Dropbox. So every time you quit Quicken, it creates a backup in a location both computers can access on Dropbox. The only manual step is that to start your next Quicken session, from either computer, you need to drag the backup file from Dropbox to your Mac desktop and double-click it to launch it. When you quit Quicken, a new backup is created in the same place. (You also have to delete the working copy from your desktop, and make sure you always grab the most recent backup file on Dropbox.)

If you don't use Dropbox, you could turn on file sharing on one computer, and then use your "Public Folder" (which exists by default in macOS) to store the .zip copy of your Quicken data file between sessions. Either computer could access the shared Public Folder on your computer to copy the .zip file to the Desktop (or Documents folder), delete the copy on the Public folder (to keep there from being multiple versions around which could cause confusion), and use the file; then quit Quicken, compress the file, copy it to the Public Folder, and delete the copy on your Desktop (or Documents folder). It sounds like a lot of things to do, but once you see the flow back and forth, you'll hopefully find it pretty easy to keep track of. Compressing a copy and copying it after using will take only a few seconds, as will copying the compressed file to the Desktop to use it the next time. 

That's a long-winded answer to a simple question, and the need to move your data file back and forth may seem a bit daunting. Once you set-up a system that works for you, and get the workflow worked out, I think you'll find you can make this work spending only a few extra seconds each time you use Quicken.