So, given that a Martian calendar will have the same weekdays, and will also have months, with numbered days within the months (be it 16x42. 23x29 or 12x56), what remains is the epoch: what year is it on Mars?
The simplest way is to have Mars year zero be the year we first land people on Mars. Alternatively, Mars year zero could be the year Mariner 4 flew by (1964), which has the advantage of already being known. I'd also suggest two others right away: we could use the same epoch as the Earth calendar does (in which this Mars year would be A.D.M. 1068), or we could use the Ecclesiastical calendar's epoch, in which this year (7517 A.M.) would be 3996 A.M.M.
I would suggest that for ecclesiastical purposes, we use the ecclesiastical epoch. Perhaps the Martian New Year should be set to be the same Martian day it was on September 1, 5510 BC (0 AM. — yes, the Byzantines didn't use zero. But if we do, it will simplify our calculations. And I'm not convinced the world was created in 5509 BC anyway.) That makes today the 495th day of 3996.
If the month lengths in a Martian day go 42-42-42-41-42-42-42-41-42-42-42-41-42-42-42-41.6 (the last month containing any leap day), you'd get a nice, regular pattern for each 167-day quarter. That puts us on the 35th day of the 12th month (3996:12:35).
Now, as for calibration of weekdays: I'm going to propose that we make Sunday, March 25, AD 31 (which would have been Orthodox Kyriopascha, had there been such a thing at the time) also have been a Sunday on Mars. Using an Excel spreadsheet (available upon request once I rectify a couple idiosyncrasies), I work out that today is Monday.
Earth's Pascha (April 6, Julian) this year falls on a Martian Thursday; I think Martian Pascha would be three days later, 3996:13:35. Martian Orthodox Lent, therefore, began on Monday, 3996:12:28, seven days ago. So their Lent is two days behind ours, at the moment.
In summary, on the Byzantino-Martian Calendar, today is Monday, the 35th of Duodecember, 3996, in the second week of Great Lent.