I have an older model kitchen faucet and the sprayer nozzle finally broke on me. I am trying to conducte a quick connect hose and new nozzle but the connector from the old sink just won't come off. I have included a picture as to what the old connector looks lick any help would be great.

If the nozzle is the only thing broken, you should be able to install a replacement spray nozzle. HD, Lowe's etc sell them. Unscrew the old nozzle at the sprayer and the bottom is normally held on with a small steel c clip. Remove the c clip, install the bottom, replace the c clip and screw the parts back together. If the hose is broken, looks like the male connector is soldered on. 1 option would be to cut the fitting off, install a compression fitting with the same diameter and screw pattern as the replacement hose. These fittings are usually soft copper so be careful bending and cutting.

Looks like that might be a push fit fitting. See the black ring on left side, and the yellow ring on the right side? I haven't seen this exact settup on fuacet spray supply, but that's what it looks like to me. So, try one side at a time. Push the pipe into the coupler while pushing the ring evenly into the coupler, then while still holding the ring in, pull the pipe out. That should do it. Let us know how things work out. Oh yeah, can you tell the us the brand of the fuacet?