B757 flight crew reported malfunction of hydraulic system in cruise flight followed by a TCAS RA.

Narrative

While level in cruise; we received a C HYD QTY EICAS message. Upon transferring the controls and radios to the FO; I ran the QRH and sent a message to Maintenance and Dispatch informing them of the message and that the C HYD was indicating .31 and decreasing. I looked at QRH in anticipation of loss of hydraulic pressure. We noted the L and C autopilot would have limitations and decided to transfer it to the Right Autopilot. I proceeded to call Maintenance and Dispatch on ZZZ radio. Maintenance decided they wanted us to perform QRH XX.X and turn off the Center Electric Hydraulic pumps as the center quantity had continued to decrease to about .1 bouncing as low as .03 at one point. We decided to advise ATC to receive priority handling to ZZZ as weather was developing. While discussing with MX (Maintenance)/Dispatch; I heard a traffic advisory followed shortly by an RA. I was in the middle of a conversation with MX/Dispatch so by the time I located the traffic and caged back to flying the airplane I heard 'cleared of conflict' and noted the FO was climbing back to FL340. I notified the FAs and then made a PA informing the passengers they may see fire trucks upon landing. We ended up receiving priority handling into ZZZ and landed successfully. We cleared the runway and taxied to the gate without further issue.