Many RAID cards support what's called "capacity expansion", oftentimes without even rebooting (in that case it's called "online capacity expansion"). You'll have to look up in your particular case whether it supports it or not. There's no inherent limitation to RAID 6 that prevents that, and I've seen cards support it. After you expand the RAID, then you expand the filesystem. In this case I think you'd do it by going into disk manager and then right-clicking the file system and choosing expand.
I'd strongly recommend ensuring you have a good backup of the RAID before trying this in case something goes wrong such as a bug with the RAID controller or you lose power mid-operation and that messes something up, etc.
Keep in mind that a local RAID is better than nothing but is not backup. What if you got a virus or there was a fire or theft. There are many cloud backup providers that will let you back up unlimited data for a small fee per yaer (Backblaze, CrashPlan, etc.), though granted that will use a lot of bandwidth if your files are large. CrashPlan is nice in that you can do an initial "seeded" backup where you mail them a drive to get your backup up to speed initially without having to transfer terabytes over your internet connection.
As a side note, if you got consumer-grade SATA drives which weren't made for a RAID, you might have issues with them dropping out of the RAID because they spend too much time on error correction and then the RAID thinks they failed. NAS-rated or enterprise-rated SATA drives are designed to prevent this.
