B737NG Captain reported a minor passenger injury resulted from a wake turbulence encounter at FL340 in ZDC airspace.

Narrative

At 1hr+02min into flight out of ZZZ bound for ZZZ1 and moments after commencing a climb from FL340 to FL380; aircraft encountered wake turbulence (we believe) from a northbound aircraft above our altitude (in fact; our climb was delayed until clear of this aircraft). Encounter was significant; causing aircraft to roll left to approximately 30 deg AOB. AP was disconnected by CA (PF) (Captain Pilot Flying) and return to straight and level flight was completed. Re-engaged AP and continued climb. I (CA) then handed controls and radios to FO (First Officer) and called back to the cabin; where I was informed that a passenger (XX_01); approximately 55 yrs old; female) who had been in the aft galley waiting to use the aft lav; had fallen during the aircraft roll and impacted the slide compartment on door 2R. Passenger indicated she had twisted her knee/leg and then went into the bathroom. Upon hearing this; I initiated a 'call me' with Dispatch to get medlink on the line and awaited further details. While awaiting status of our passenger; myself; my FO and our jumpseater collected flight parameter details. As noted; 30 deg left AOB (Angle of Bank) occurred; AP disconnected. Of note: no overspeed was experienced; no 1g was felt; and no 'bank angle; bank angle' audio was heard. When referenced; did not 'drop a turbulence pellet'. Entire sequence lasted maybe 3 seconds and no turbulence was called for by ATC; noted on WSI; [display] or FD Pro High Level views. FA (Flight Attendant) got back to me and informed me that passenger was in pain but had made it back to her seat. She was in a center seat but we had her reseated to an aisle seat for comfort. I asked FA to ask passenger if she would like to have aircraft met by EMTs (Yes) and/or did she want us to divert (No) given approximately 1hr of flight time remaining to ZZZ1. Meanwhile; got on the radio with Medlink and gave all the details. Dr. advised ice pack and 3 acetaminophen (aka Tylenol) from the kit be administered. Passenger refused the Tylenol and a cold pack had already been applied by the FAs. Made a passenger announcement somewhere in here letting folks know that we had encountered an aircraft wake; and that the aircraft was fine and we would be proceeding to our destination. Coordinated with Dispatch to have aircraft met by EMTs with an aisle chair; and backed this up with an ACARS msg to Station. Informed Center and ZZZ1 Approach that we would like an approach to Runway XXR as it expedited taxi to the gate. Approach; landing; and taxi in were uneventful and all other passengers were asked to (and complied) remain seated until this passenger was removed by EMTs.