Commercial Fixed Wing
Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Flight Crew Last 90 Days 115; Flight Crew Total 1835; Flight Crew Type 1835
Situational Awareness; Confusion; Time Pressure; Communication Breakdown
Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued New Clearance; Air Traffic Control Issued Advisory / Alert; Air Traffic Control Provided Assistance; Flight Crew Returned To Clearance; Flight Crew Became Reoriented
Chart Or Publication; Environment - Non Weather Related; Human Factors
Good VFR day; cleared departure out of ZZZ to TEHRU; then direct MTJ. Just prior to TEHRU we informed the DEN Center controller that we would be requesting the ILS [Runway] 17 and was expecting to be vectored to the localizer. Instead; we were given 'direct DIRDY; Cross DIRDY at or above 13;000 feet; cleared for the ILS 17.'Loaded that into the MCDU; which inserted the procedure turn depicted to the north; after BRUUK. As the track from GEJYU to BRUUK is 78 degrees off the localizer course we figured this was the only 'legal' way to fly it; based on our clearance; so wasn't surprised. (Coming from the west side of the localizer there is a DME arc with a lead-in radial depicted to account for that steep angle; but not from the east. Again; we assumed the procedure turn was the only legal way to turn onto the localizer.)Although the crossing altitude for GEJYU is 10;800 feet the initial procedure turn altitude is 12;500 feet so we crossed GEJYU somewhere above 13;000 feet on our descent to 12;500 feet. Unfortunately; when we visually picked up the airport this made us too high to ask for a visual approach so we decided to just fly the procedure turn; as planned.After becoming established in the procedure turn outbound; the Controller asked us what we were doing and said most flights just join the localizer and go directly toward the runway. After some back-and-forth about this; while we were in the turn back toward BRUUK; he said 'low altitude alert; climb and maintain 13;000 [feet]' (we were still at 12;500 [feet]; and VFR). We climbed to 13;000 feet; called the runway in sight; and were cleared for the visual approach.Long story short(er); there was obvious confusion on our part regarding how this approach was supposed to be flown; as well as; I believe; based on our conversation; the Controller's. After landing we looked over the chart for 5-10 minutes; looking at all the various notes; etc.; before finally noticing the 'NoPT' note depicted under the 9;000 feet MEA listed for the leg from GEJYU to BRUUK.I saw the altitude and NoPT but when looking at the routing in the FMS and the procedure turn altitude assumed it wasn't intended for flying the full approach. I also don't understand why we would expect to intercept a localizer from a 78-degree angle when; I believe; while being vectored to an intercept; the intercept heading given by ATC is required to be within 30 degrees of the localizer course. If we had flown the depicted track from GEJYU to BRUUK as apparently expected on this approach; I believe the airplane would have overshot the turn onto the localizer by at least 1-2 miles. As for the low altitude warning the airplane was flying the published procedure turn on altitude which would lead one to believe you were in protected airspace?
Air Carrier First Officer reported they misread the approach chart and did not fly the approach track correctly into MTJ airport. The crew was given a 'Low Altitude Alert' by ATC.