CE700 First Officer reported failure to properly program the FMS along with CRM communication confusion resulted in an ATC low altitude alert and a CFTT event.

Narrative

During the briefing before our flight the Captain (PF) briefed that he would be doing the noise abatement procedure. He asked that I (PM) set a VS of 1500 FPM after takeoff and gear retraction. On departure from Runway XX at ZZZ we began climbing with the autopilot off and the autothrottles on; on the ZZZZZ. After gear retraction; I selected VS and set 1500 FT as the Captain requested. As we were climbing past ZZZZZ1 and towards ZZZZZ2; which has a mandatory altitude of 1500 FT - I estimated that at our current rate of climb; we would not make this 1500 ft restriction at ZZZZZ2 and I became concerned. I do not recall if the Captain was completely pitched into the flight director or not during this time; I just knew we weren't climbing fast enough.Once I noticed we wouldn't make the altitude at ZZZZZ2 - I told the Captain verbally 'We need to climb' and immediately followed up with 'Climb'. Just after I said this; the Pitch Limit Indication (PLI) came into view; and the Captain pitched down to avoid touching the PLI. We briefly incurred a -100 FPM decent rate due to this pitch change. I knew that due to the PLI being in view we needed to abandon the noise abatement; but I didn't communicate that to him in the moment:I again re-iterated that we needed to climb; but while I was saying this I was interrupted by ZZZ Approach over the radio who said 'Aircraft X; low altitude alert climb immediately' and I responded to that radio call. We were about 1200 feet defending to 1100 feet when ZZZ Approach warned us of our altitude. Around the time I was responding to the radio call; the Captain pitched the airplane higher and also selected a higher VS. Due to this we ended up significantly increasing our climb rate and actually arrived at ZZZZZ2 at the correct altitude of 1500 ft. The Captain then had me engage the autopilot and he moved the SPD knob to FMS and we climbed out as normal without further incident.Day 0 XA:54 ZCorrection to above statement 'After gear retraction; I selected VS and set 1500FPM' not '1500 FT.'Day 0 XC:27 ZAdded edit for clarification that when I verbally said 'climb' and 'we need to climb' I did not get a verbal response from the Captain that he understood me and was going to add power and climb. However in the moment I believed he had heard and understood me but was too task saturated flying the airplane to respond and acknowledge me. Suggestions: Whenever the PLI comes into view; according to the Company AOM the noise abatement procedure should be discontinued. Upon reflection; while I told the Captain to 'climb' and 'climb immediately' I was assuming this would cause him to abandon the noise abatement procedure. I assumed that my call to 'climb' would cause him to turn the autothrottles off and advance the throttles manually but it didn't have the desired affect and I should have been more clear with my callouts. I should have told him 'abandon noise abatement; add power.'Additionally after reflection on the incident I realized that the FMS was incorrectly set for this noise abatement procedure. In our case the SPD knob was set to Manual (MAN) and 180 knots was set in the window. Our AOM directs the SPD knob to be in FMS; with the FMS speed set to 180 along with 3000 ft and 5 miles as parameters. I did not catch the error in programming the FMS during our briefing and subsequent tax - and I should have. I would also suggest that the noise abatement procedure be focused on in recurrent trainings; line checks and otherwise to ensure all crews know how to correctly and safely execute it and when to abandon the procedure (PLI in view).