Air carrier flight crew reported they were cleared for initial approach and noticed a discrepancy between the FMS glide path and the PAPI glide path. After they landed; they realized the altimeter setting was incorrect.

Narrative

Cleared for RNAV X 31 into LGA; aircraft configured and stabilized IAW (In Accordance With) with company RNAV X 31 pages; aircraft configured at 1000ft stabilized. 500' aircraft configured and stabilized. It was unknown to the First Officer and myself that the aircraft was high on the approach until rollout to align the aircraft with the runway after the left hand turn base to final on the RNAV X 31 approach. The First Officer disconnected the autopilot around 1000' and continued the approach following the FMS without any lateral or vertical deviation. As the aircraft was maneuvered to align with Runway 31 it was noticed that we were high according to the PAPI. However; according to the FMS the aircraft was right on glide path. We were unsure of the issue at the moment; the First Officer disregarded the FMS glide path and decided to increase descent to align with the PAPI vertical path. When doing so the aircraft was in a consistent 1000' per minute descent briefly reaching 1200' FPM with an immediate correction back to 1000'. We both knew something was not right considering the discrepancy between the PAPI vertical glide path and the FMS vertical glide path. Unfortunately with the large left hand turn the PAPI are out of sight for a majority of the approach; decreasing our situational awareness of our actual glide path. The First Officer and I both were prepared to call for a go-around; however the aircraft was still within the stabilized approach criteria and given the visual conditions and the current descent rate it was in both of our judgement that a safe landing within the touchdown zone could be made. Not leaving out the possibility that a go around could still be called if the circumstances changed. The aircraft touched touchdown within the touchdown zone and a full stop was made without any further issues. The question still remained. What happened? Why did the FMS have a large discrepancy when compared to the PAPI? After parking the aircraft at the gate a long debrief commenced without a solid answer. It was not until retrieving a new ATIS that I noticed that the airport altimeter 30.21 and we had 30.10 set. It is still unknown where we received the altimeter setting of 30.10. My assumption is that when cleared to descend VIA the STAR that 30.10 may have been given; or when checking in with approach an altimeter setting given by the controller was misinterpreted by us or misread by the controller. Again these are assumptions and cannot conclude the actual reason for setting 30.10. A 30.10 setting did not seem unusual given that we began the day with an altimeter setting of 30.40+ and had been decreasing rapidly throughout the day. To further solidify the importance of a correct altimeter setting it is in my opinion that if this approach was conducted in IMC; once clear of clouds there would not be enough time to maintain a stabilized approach to land and an immediate go -around would be conducted.Suggestions:This is a valuable lesson learned when given an altimeter setting to double check it with the airport digital ATIS or ask for clarification from ATC if unsure.