Can I sync Quicken on my MacBook Air and Mac desktop?

Quicken isn't really designed to do that, and if you try it with the Quicken cloud syncing functionality, you may find data getting out-of-sync or being overwritten.

Quicken is built around there being one data file on one Mac. You can use the cloud sync services -- the Quicken mobile app and/or the Quicken web login -- to enter and see some of your data online on the other Mac, but only one Mac contains your data file and has all the features of the desktop application. This cloud sync was not built to support moving data between data files on two different Macs. you may be able to get it work work, but I wouldn't recommend it. Using the web interface to your Quicken cloud data is probably a better way to go if you just want to see or enter some data from your secondary Mac.

The other alternative is to move the Quicken data file between the two machines -- via flash drive or via a cloud service like Dropbox or iCloud. This is safe as long as you move a compressed version of the file (either File > Compress in the finder, or Save a Backup in Quicken). Do not try placing your live Quicken data file on a cloud storage site and accessing it there; it's not supported, and file problems often occur. But if you pass back and forth a compressed or backup version of the file from one Mac to the other, and long as you keep track of which Mac has the most current file, and are sure to pass it back to the other Mac, that's a two-Mac solution some users have found workable.