No, you cannot install 2 different versions - Windows can only have one NBU Client Service and Unix can only have one bpcd and vnetd 'listening'.
NBU 6.5 can backup NBU 5.1 clients.
Whether it's advisable to backup one client to two different environments is another debate... You will only be able to backup databases and ALL_LOCAL_DRIVES to the environment that is listed as MASTER server. Normal filesystem backups where the folders are listed explicitly can be backed up to two environments as long as you have all the necessary SERVER entries on the client.
Restores initiated from the client will by default be directed to its defined MASTER server.