No you wouldn't, for two reasons:
1. The performance of your SATA III hard drive is determined by the specification of the SATA ports on your computer's motherboard. If the ports are SATA II, a SATA III drive's performance will be limited to SATA II
2. There's no such thing as a SATA III cable. It's just a marketing ploy to charge more money for them. All SATA cables are the same as regards SATA specification, they are just SATA cables. The manufacturing quality may vary across different brands, but that's all.