Hi,
Don't really know much of acQuire but googling seems to indicate it is possible to connect to the database through ODBC (http://help.leapfrog3d.com/Geo/4.1/en-GB/Content/drillholes/connect-to-acquire-db.htm?Highlight=acQuire#). It is very possible to read data with ODBC in FME if you have the correct drivers. (Maybe acQuire do use a known/standard DB and then you probably could connect with a predefined Reader instead).
Then if the database-schema is somewhat understandable you would very likely be able to read table-data. It can be tricky to handle geometries if that is stored in some custom way - but it can also be very simple depending on how it is stored.
I would give it a go.
@ejdkeys if @ulfme's suggestion doesn't work out for you then you may want to check this out - https://www.acquire.com.au/download/fme-plugins-for-acquire-4-5-4-4/ You might have some luck asking acQuire about this. I will do some investigating internally to find out some more information on this end for you as that link looks a little dated.