There are two ways that I can think of. The first is to grep the JAMFSoftwareServer.log. The version is written on every startup. The other is to check the version.xml inside the deployed application. It should be in $JSS_INSTALL/tomcat/webapps/ROOT/WEB-INF/xml. It holds all of the version of different binaries in a normal xml format.
I guess you could check the database for the version, but I haven't looked. You would think you could get this info pretty easy from the api, but its not there..
FWIW, had to do this today on a Mac server running 10.1.1...yes, I know this is a Linux thread but figured someone searching might find this useful:
# xmllint -format /Library/JSS/Tomcat/webapps/ROOT/WEB-INF/xml/version.xml | grep webApplicationVersion | cut -f2 -d ">" | cut -f1 -d "<" 10.1.1-t1513360285
I like the approach suggested by @donmontalvo better. This works even if the JSS webapp is not running or broken. It does however require you to have terminal access to the JSS server e.g. via SSH.
For the benefit of Linux server users due to the different file path the command would be -
xmllint -format /usr/local/jss/tomcat/webapps/ROOT/WEB-INF/xml/version.xml | grep webApplicationVersion | cut -f2 -d ">" | cut -f1 -d "<"
If your servers are working successfully then the approach suggested by @daworley is simpler.