How do I know if my MacBook Pro supports SATA III?

You can take a look at the Link Speed under the Intel 5 Series Chipset section. SATA levels are named after a number (1, 2, 3) and also for their speeds.

• SATA 1.0 is known as SATA 1.5Gb/s
• SATA 2.0 is known as SATA 3.0Gb/s === Your computer supports SATA 2.0
• SATA 3.0 is known as SATA 6.0Gb/s

"Link speeds are the theoretical maximum speed at which data can transfer to or from those devices." - Taken from here.

The "Negotiated Link Speed" is what you currently have, in your case, this is SATA 2.0 too…

Here is a link with a description of the different SATA revisions

It really supports any SATA version I, II, or III. The SATA drives and are backwards compatible. If the drive is SATA III and your MB Pro is SATA II, the dive will perform at the theoretical levels of a SATA II drive. Same for SATA I.