I should convert an ArcGIS cartography style (where the Esri fonts were used for the symbology) in QGIS. Is it possible to use the esri symbolic fonts and why some of them (eg ESRI Oil, Gas, & Water) are not correctly detected by other software?
Provided they are true type fonts and provided that you don't in some way violate ESRI's licence terms by using them then you can include them in a QGIS style.
You can not convert ESRI Style to be used in QGIS, because The ESRI Style are in binary encoded format, and they have not disclosed their inner structure, the only format that you can convert them directly is to .style format that is not supported by QGIS.
One way you can use Arc2Earth, to convert it into SLD format, but this is not free now.
For windows, you can also use arc2sld converter, to convert it to SLD. SLD generated from this does not incoporporate high level design.
One other approach that you can use, is you can create your own vector graphics, and then import them into XML, and then import them directly in QGIS, as QGIS support xml format to import symbology.