Light jet Captain reported they received a low altitude alert from ATC departing SJC.

Narrative

Departure events from SJC.Weather: winds 310/3 CLR 13/4 3016 RW in use 30L and 30R. Aircraft configuration: Flaps 15. Take off trip. ATT. Using LNV for departure. Departure clearance: SPTNS1.TECKY VALREE. Climb by SID to 5000 ft. by ATC. Crew inputs on the FMS SID; flight plan and review at the ramp. Taxi Clearance: taxi G; W; hold short of 30L on B. Crew follow taxi instructions taxi to holding point B and RW 30L. Short after holding short clearance was issued to cross RW30L; back track RW30R full length for departure. Crew programs new assigned runway and SID on the FMS. Take off clearance given before reaching end of 30R. Take off runway confirm; aircraft configuration. Crew performs a normal take off.Initial Climb: Gear up at positive rate and confirmed LNV. Flaps up on speed 145 kts. FMS indicated right turn at 900 ft. (RA indicated 880). Crew follow flight director right turn south bound to SPTNS. Crew notice FMS did not sequence properly to over fly MLPTS before commanding right turn. Crew increases climb rate and shallows bank and changes frequency to Departure. ATC informs crew they received a 'low altitude' warning and confirm early south turn. ATC advises crew to expedite climb to new altitude of 12;000 ft. Crew kept aircraft under full control to prevent CFIT or any terrain warnings. Crew have learned from this event not to rush. Not to accept a takeoff clearance without making sure all is in place and to always confirm FMS sequence properly. Furthermore I am enrolling on a CRM refreshment course and airport terminal area procedures to include STARs and SIDs.