Hawker 800 pilot reported receiving an altitude alert from ATC during the approach although the flight crew reportedly was crossing the fix at the correct altitude.

Narrative

In the early morning (approximately XA:00) of the day in question; my Captain and I were dispatched for a medical flight. After taking the medical team to their destination (ZZZ1) and while waiting for the team to return; we were trying to get some rest at the FBO. We both got some rest and determined we were rested enough and felt safe to conduct the flight. The Captain did not receive the notice when the medical team was on the way. When I received a text message that they had arrived; we rushed to the plane. The Captain directed me to file a flight plan and; due to the lack of time and the fact that he had checked the weather; refused to discuss it with me at that time; despite the threat of thunderstorms. All of this led to a stressful cockpit. We departed to our destination (ZZZ) at approximately XJ:20. We encountered thunderstorms throughout the route; which we overcame by deviating off course. We started our descent with Center Control; who asked us to descend to an altitude of 12;000 on the way to the ZZZZZ waypoint (the first waypoint on the RNAV approach). Due to our close proximity to the waypoint; this caused a steep descent. Then the Center transferred us to the Approach Control. While we descended towards the RNAV approach; the Approach Controller cleared us for the approach and asked us to cross ZZZZZ waypoint at an altitude of 10;000; which caused an even steeper descent. After crossing ZZZZZ; we continued the descent according to the approach plate. At the ZZZZZ1 waypoint; we descended to 5;300. While we were descending through the approach plate; the controller advised us on a low altitude alert. The Captain advised the traffic controller that we are at the correct altitude according to the approach plate. We continued the approach and landed with no problems. At the brief after the flight; the Captain and I both raised our concerns; which included the environment of the cockpit and the fact that we were not alerted to the medical team's arrival with enough time to adequately prepare for the flight. As I am a new pilot on this jet and am still learning; I had a lot of trust in the Captain; but I wish he had communicated better at time of the event. I have learned that it is important to not just blindly follow directions of the traffic controller. For example; when directions will result in a steep descent; I should consider if we are safely and comfortably able to do so. I have also learned that a stressful cockpit leads to lack of good communication and possible safety concerns.