Air carrier Captain reported they deviated slightly from the SID due to a discrepancy between the Captain's and First Officer's Flight Director and Navigation Display.

Narrative

Departing ZZZ XXR; on the ZZZZZ SID. On initial climb PF (Pilot flying); started to turn prior to ZZZZZ FLYOVER pt. PM (Pilot Monitoring) immediately corrected PF and directed PF to fly direct to ZZZZZ. PF started turn and was showing on FD (Flight director) and zero deviation on NAV route as displayed on his ND (Navigation Display). After 5 seconds into turn PM noticed that they were 0.2 right of the NAV line and directed PF to correct back to NAV. PF responded that he was right on as his FD and ND indicated. Departure called and asked if we were flying departure. We indicated we were but we're having a discrepancy in our NAV indications. The controller gave us a heading of 140; which was a continuation of our right turn we were currently performing. After what was less than a minute he told us to continue our turn to ZZZZZ1 fix and continue as filed. On initial climb we were assigned 5;000 ft. Upon contacting departure we were given a climb and maintain 15;000 ft. At no time did we exceed max airspeed of 230 knots. The climb and maintain 15;000 ft was assigned prior to above noted discrepancy. The most the PM showed off was 0.2 DME from what the PM showed was on course; or zero deviation. Discrepancy in Nav systems. PF's system had to be in error as compared to PM; as indicated by departures query. The discovery of differences and query happened within seconds of each other. Discrepancy was noted almost immediately. The PF was flying his FD; and was showing on course for his NAV as displayed on ND. Event was quickly resolved with aid from departure controller.