Date: 202107
Local time of day: 1201-1800
Location: ZZZ.ARTCC, US
Aircraft: Commercial Fixed Wing
Operating under: Part 121
Flight plan: IFR
Mission: Passenger
Flight phase: Climb
Airspace: Class E ZZZ
ATC / advisory: Center ZZZ
Second aircraft: Any Unknown or Unlisted Aircraft Manufacturer
Reporter organization: Government
Reporter function: Enroute
Reporter qualification: Air Traffic Control Fully Certified
Human factors: Confusion; Distraction; Situational Awareness; Human-Machine Interface
Second reporter function: Pilot Flying
Anomaly: ATC Issue All Types; Conflict NMAC; Ground Event / Encounter Ground Equipment Issue
Detector: Automation Aircraft RA; Person Flight Crew
When detected: In-flight
Contributing factors: Airspace Structure; Human Factors; Procedure; ATC Equipment / Nav Facility / Buildings
Primary problem: Procedure

Narrative: Approach flashed (handed off) Aircraft X departing ZZZ going to ZZZ1 to me. The aircraft was out of 8;500 feet when I took the handoff and checked on out of about 9;400 feet climbing to 16;000 feet. I climbed the aircraft to 2;6000 feet as I had no traffic in the area. The majority of my attention was focused on the south side of the sector where I had a VFR aircraft inbound to ZZZ2; an IFR aircraft inbound to ZZZ2 who was just out of 40;000 feet 45 miles from the airport as well as an additional VFR aircraft who was climbing and descending that needed lost communication procedures. Shortly after climbing Aircraft X; he reported that he needed to level off at 100 feet due to traffic above at 10;500 feet; I looked and I did see the numbers XXX on the scope but the target associated with these numbers was obscured by the target from Aircraft X.I was trying to figure out if there was actually traffic there and if there was traffic there why I had not received any kind of CA (Conflict Alert) indication (nothing was flashing at all) and since this was well within the lateral and vertical confines of Approach's airspace why they had climbed the aircraft into traffic within their airspace. I acknowledged Aircraft X and approved his request to level off; at that point he told me he was responding to an RA. I was surprised by this; but just asked him to report on course and when he resumed his climb. A few minutes later he told me that he was going to report the incident. At that time I responded trying to calm him down and let him know that I had already notified the supervisor that he had responded to an RA. A few minutes after that; Aircraft X pilot informed me that he was going to report it as a 'near midair'. Again I attempted to defuse this situation; informed the Controller in Charge and told him I would contact Approach to find out more details. We then gave him the information to contact the Center Operations Manager in Charge as well as the Approach Control Supervisor. Approach Control should have called the traffic and not shipped me the aircraft until they were clear. The Conflict alert should have been going off for several minutes; which would have alerted both myself and Approach Control to the unusual situation. At that point I would have seen the non-associated VFR target and called the traffic. We should have better staffing so that we aren't working 6 day work weeks and running 2 under on busy shifts. Everyone is burnt out; exhausted and probably not as sharp as they would be.


Second narrative: On departure out of ZZZ;  we received a RA after encountering a near mid-air collision. The event occurred around around 30 to 40 miles south of ZZZ. The Captain was Pilot Monitoring and the First Officer was Pilot Flying. The First Officer was hand flying at the time of the event. We were climbing to 16;000 feet as instructed and talking to Departure Control. 16;000 feet was our initial altitude assigned on the ground in ZZZ. We received a handoff climbing through approximately 8;000 feet and contacted Center. After we checked in with Center; I noticed traffic straight ahead at 10;700 feet depicted on the scope and we were climbing toward them. We were not notified by ATC of any traffic and once myself (First Officer/Pilot Flying) noticed the traffic at 5 miles; I called 'traffic 5 miles 10;700; I'm going to level off' and began to level off immediately around 9;000 feet and did not climb above 10;000 feet.We got a TA; and I began a slight left turn as I noticed on TCAS that traffic was crossing left to right of our course. As the Captain was calling Center to ask about traffic in the area we got a visual on the traffic and immediately started a descending left turn to avoid traffic; simultaneously we received a descending RA and complied. The converging aircraft passed off our right wing and appeared to make a left turn also to avoid our converging course. We never saw the converging aircraft change altitude more than a few hundred feet. We notified Center of the RA event and returned to normal flight. Prior to departure; we briefed General Aviation traffic as the biggest threat in the departure briefing and I think that helped our awareness.General Aviation traffic not on a IFR flight plan or talking to ATC. Departure or Center not notifying of traffic. Timing of handoff from Departure to Center occurring at the event. Suggest attentiveness to traffic.


Synopsis: A Center Controller and air carrier flight crew on Center frequency but still within the confines of the Approach Control airspace reported a NMAC with an unidentified VFR aircraft. The Center Controller's Conflict Alert warning did not activate.