This answer claims that enquire whether differs from enquire about whether. So what does the latter mean?
Here are some indications that enquire about whether is correct usage of some sort.
Source: p 4, Pragmatics and the English Language, By Jonathan Culpeper Ph.D in Linguistics (Lancaster Uni), Associate Prof in Linguistics Michael Haugh PhD (University of Queensland)
What this joke illustrates is that the whole utterance Do you have any firearms with you? can have more than one meaning: is it an enquiry about whether the driver has firearms or a request for firearms?
Source: p 70, The Phraseology of Administrative French,
by Wendy Anderson PhD in French linguistics (University of St Andrews)
In response to a query about whether a corpus of written academic prose in the disciplines of commerce/economics and natural science/history should be called specific registers or genres, the four linguists who commented all proposed different solutions, ... .
Here's another quote by Prof of Linguistics, Frederick Newmeyer.
All that said, what distinguishes enquire whether and enquire about whether?