As Daxlerod has stated you should use the SMART tools, however you also need to perform what is called data scrubbing.
If you haven't already, create a disk group. For data scrubbing to be available you need at least 3 disks, with at least 1 redundant drive and at least 2 data drives. Wait for the disk group to fully synchronise, then perform a data scrub.
As of Synology OS v4.2 data scrubbing can be accessed from
Storage Manager -> Disk Group -> Manage -> Start data scrubbing
This will take hours as it reads all sectors of all of the disks and performs some math to see if the checksum data adds up properly. You can use you NAS while this is going but it will be a bit slower. Many people run a data scrub once a month.
There is a good article called Scrub Synology RAID disks that also tells you how to run the file system check tool called fsck even though there is no option in the web interface for it.