You can change, yes. BUT you will NOT get a speed-up in performance.
I am assuming you propose to get a new HDD of the classic mechanical kind - that is, with spinning disks and moving arms, etc. You are NOT getting a SSD - "Solid State Drive" - right?
With ALL mechanical drives, the limit on performance is the time it takes for the mechanical components to move - the rotational speed of the disks on the shaft, and the speed of moving heads on an arm to the correct track. It turns out these processes are MUCH slower than the maximum speed of the data transmission system between the HDD's own controller board and the mobo controller it connects to. It is that last Max communication speed that is the "label" on SATA drives - SATA 1.5 Gb/s, SATA 3.0 Gb/s (or "SATA II"), or SATA 6.0 Gb/s ("Sata III"). Moreover, that mechanical drive limit turns out to be close to the oldest and slowest communication system. Typical actual average data throughput on a SATA drive is around 100 to 150 MB/s; some of the fastest newest ones can get a little over 150 MB/s, but none currently even come close to the 300 MB/s speed that is a "SATA II" unit's MAX communication speed. The only units that can meet that (and may exceed it to get closer to "SATA III" speeds in future designs) are the SSD "drives" that do not have mechanically-moving parts.
So, unless your old "SATA II" drive is one of the slower ones under 100 MB/s, and your newer one is at the upper end over 150 MB/s, you will not see a performance improvement. It is VERY unlikely that you would see a doubling of performance. This is the reason that I suggest that, if you're sticking with classic mechanical HDD's, it is not worth paying extra $ for SATA 6.0 over SATA 3.0.