beta sp and sync generator
 

I have an AJA Io HD, connected via component cables to a Sony PVW-2800 Betcam SP also a Sony DVR-20 D-2 Composite Digital, and a mac pro (quad-core). I am capturing to FCP, with the setting Apple ProRes 422 but the video is very low quality and has color blocks. I am also having problems with tracking/calibration.

How would I fix this? Would I use I synchronizer? And are a synchronizer and time base corrector the same thing? Or is there a synchronizer included in my sony Beta SP deck?
I have an old Leitch Sync Pulse Generator 1300N. Would this do the trick? If so, how would I set it up?

How does the playback image look on a regular monitor connected directly to the decks?
Have you tried a head-cleaning tape if the image looks bad when viewed directly from the deck?
The PVW-2800 has a TBC built-in. You should use the blackburst output from the sync generator to syncronize both the deck and the AJA, making sure the blackburst signal is terminated correctly.
There are also some configuration choices to make so that all parts (deck, AJA, FCP) are expecting the same form of component signal.
Remember that the original tapes will not be high def material, so you have a limited amount of image quality to work with.

The playback image is great on the monitor. There is no distortion or pixelation.
Which output is the blackburst?

I'm having trouble locating a manual or image of the connectors on the Leitch 1300N. Can you list the names of the connectors and I'll tell you which one it is.

Yeah, I haven't been able to find any information on the 1300N either.

The connectors on the back are listed as: BLK, Test Signal, S/C, Sync, BLKG, H DR, V DR, Burst Flag, Frame Ident, Lock Input.

You should use "BLK".
You could also use the Test Signal connector if you can select something normal like colorbars. It doesn't have to actually be black, it just needs to be a stable composite video signal.
The PVW-2800's reference input connector is a "loop-through" with two connectors. You need to set the termination switch beside those connectors depending on whether you are looping through with two cables or ending the connection there with just one cable.