No, the SY-300 doesn't convert pitch to MIDI, nor can it send or receive MIDI note data. At least, not according to its MIDI spec sheet and what Boss have said.
I looked into one a while ago hoping it was a cheaper alternative to the Roland synths, which are full of stuff I have no need for, like modelled guitars, fx and amps, while what I want to be able to do is just use a guitar to control the real synths I already have.
I've never worked out why Roland don't offer a basic MIDI interface that matches their guitar pickup, has inputs for more than one control and sustain pedals and simply takes the signal from theor MIDI pickup as input and outputs it as MIDI over USB and 5-pin connectors. I suspect they might sell quite a lot of them if the price was right.
Jam Origin's guitar to MIDI software is worth a look, though it's a bit of a work in progress still the basic functions are pretty good.
http://www.jamorigin.com/products/midi-guitar/