First you must determine if the coordinates are in latitude-longitude, a geographic coordinate system (NAD27) or whether they are in a projected coordinate system. Coordinates in latitude-longitude for Mexico will be between -117 and -87 degrees longitude and 14 and 33 degrees latitude.
If the coordinates are larger than that, the data's in a projected coordinate system and you may need to ask the collaborator for more information.
One issue you may run into with QGIS or other software packages that use PROJ.4 (AKA PROJ) is that a geographic coordinate system (also called a datum) is "bound" with a particular transformation and parameters. The default will likely be for the US, not for Mexico. I did find statements in this document, "Obtención de coordenadas con GPS en ITRF y su relación con WGS84 y NAD27", here that there is no general NAD27 transformation. DMA/NIMA/NGA did publish one but its accuracy is only several meters. The parameters for the transformation would be given in a PROJ.4 string as:
INEGI has an online converter here. So you can check any results you get elsewhere.
I can find references to a Lambert conformal conic-based projected coordinate system, but it doesn't use NAD27. Instead ITRF2008 or ITRF92 (both are similar to WGS84). The parameters are:
Latitude of false origin: 12°N
Longitude of false origin: 102°W
Latitude of 1st standard parallel: 17°30'N
Latitude of 2nd standard parallel: 29°30'N
Easting at false origin: 2500000 m
Northing at false origin: 0 m
Edited to add information on the LCC projected coordinate system.