Air taxi pilot reported they descended to cross a fix at 13;000 feet as instructed by the Center Controller. A subsequent controller issued a low altitude alert and climb clearance.

Narrative

In the descent phase of the flight; the Denver Center controller asked us what approach we would like into Montrose; CO (MTJ). We had already programed the FMS for the RNAV 35 Approach based on the winds so that that is the approach we requested. We were around 25000 feet on top and the controller cleared us direct to YARUB (IAF); cross at 13;000 feet. Both crew members had already briefed the approach; and missed approach procedures plus any and all threats in accordance with Threat Error Management. In addition both crew members had Terrain selected on the respective MFD's (Multi-function Flight Display) both Captains and First Officers side. The EGPWS (Enhanced Ground Proximity Warning System) in the descent was not painting any significant terrain features or painting any terrain at all. We both were keeping a close eye on this. We entered the clouds around 15000 feet and continued on toward YARUB for the crossing restriction.At approximately 13500 feet and 1 minute from crossing YARUB; a Denver Controller said they were receiving a Low Altitude Alert and directed us to climb to 16000 feet. We immediately initiated the climb passing over YARUB in the turn to the next fix which was COQKU (IF). The controller asked us what our last clearance was and we told them it was the previous controller had cleared us to YARUB cross at 13000 feet. We arrived over COQKU at 16000 feet when the controller cleared us for the the RNAV 35 Approach into MTJ. We began the descent in an attempt to capture the Glide Path and broke out of the clouds. On the approach COQKU need to be crossed at 12000 feet. Yet we were cleared for the approach almost 4000 above that. We began an descent hoping to be stabilized before EBIHE (FAF). Around 13000 feet we broke into VFR conditions. With the runway in sight we continued the descent until EBIHE approximately 6 miles from landing. There were other small aircraft in the vicinity departing and landing Runway 31 we were still attempting to get stabilized and it became apparent that a missed approach was the only option so we initiated a missed approach; contacted the ATC and asked for vectors around for another approach. This effectively put us back in a line of aircraft coming into MTJ. We became #4 in the line received vectors and landed without incident.