Air taxi Captain reported ATC issuance of a low altitude warning for the flight. Captain reported no terrain warnings and continued climb.

Narrative

We filed and received an IFR clearance for SSTIK 5 RNAV departure from SFO.Upon taxi out from FBO; we accepted the Ground Control ATC offer to depart from SFO Runway 28R at Echo intersection. The First Officer was the flying pilot and upon initial climb-out he engaged the aircraft Autopilot and selected the NAV function for our Universal NAV unit. The aircraft began to turn left towards the PORTE fix. Departure Control advised us that they were receiving a low altitude warning for local terrain as we continued our climb in visual meteorological conditions. We did not experience any terrain warnings with our aircraft GPWS system and continued with normal climb sequence.I spoke with Northern Cal TRACON. I told them that the only thing we recalled hearing from the Tower concerning our instructions to depart were 'Cleared for takeoff Runway 28R at Echo;' which I read back to the SFO Tower Controller.The First Officer and I discussed the situation and believe there was shared responsibility that was not met with ATC as well as us. Tower ATC apparently had a different departure strip that showed we were to follow the WESLA 5 RNAV departure; which is used for SFO Runways 28L/R. TRACON said they had the strip for the SSTIK 5 RNAV departure; which is used for SFO Runways 1L/1R. With the last minute departure runway change; I failed to notice the associated runway change that needed to be reprogrammed in our Universal NAV unit. Ground and Tower ATC never mentioned a change of departure procedure to us prior to clearance for takeoff.To prevent recurrence; I will double check that any assigned takeoff runways match the charted departure procedure runways and are appropriately loaded into our NAV units. If I do not feel comfortable with a last minute departure runway change; I will either stop the taxi to reprogram the NAV units or I will not accept the change. Also; I will query ATC if I believe there is any lack of clarity concerning their instructions.