Air taxi pilot reported on departure ATC issued a low altitude alert resulting in the pilot increasing their climb rate.

Narrative

On departure out of ZZZ on the the ZZZZZ departure on the afternoon of Day 0; ZZZ Departure gave me a low altitude alert advising me to climb; I responded that I was climbing and increased my climb rate in response to the controllers concern.Before departure I loaded the ZZZZZ departure into the FMS; briefed the departure; and made sure to go over all altitudes and fixes along the route as I hadn't flown the departure previously. I then verified the fixes and altitudes in the FMS and then the depicted route on the MFD (Multi-function Flight Display). I double checked and verified all this to what I thought was correct to make sure the Flight Director and Autopilot would follow the correct route being that I was flying single pilot and task saturation would be high. Upon taking the runway I selected Heading Mode and armed Nav mode on the Flight Director/Autopilot Control Panel. On initial climb after retracting the landing gear and climbing straight ahead on a heading of 306 the FMS sequenced and the Flight Director changed to the previously armed Nav mode to intercept the ZZZ [VOR] radial to ZZZZZ. Thinking back; it did seem a bit early being that the turn is supposed to be made at ZZZ DME 1.8 but I was task saturated during initial climb flying the aircraft; cleaning things up; and getting handed off to departure. I believe the FMS sequencing early for some reason caused the low altitude alert for some reason. I'm not sure how this could be corrected as I verified what I thought to be all correct fixes and altitudes.