How to quickly obtain the number of records in a persistent class?

Not to my knowledge.  While there is a global node in the storage map that is used to get the next available Id this would only work on tables/objects based on a single integer id.  At the same time, this is the next available Id and does not account for physical deletes that may have occurred, ie the next Id might be = 101 but you may have fewer than 100 rows/objects as some may have been deleted.  The simplest way to accomplish this would then to perform a SELECT COUNT(*) FROM TableName.  If the table implements this bitmap indices this should be ms.  If you don't get the performance you want you might consider adding %PARALLEL to the FROM clause and let the optimizer decide if it makes sense to split the job.

If records are never deleted and you're okay with a few misses, use ExtentSizeFast, if you need the fastest precise runtime implementation use ExtentSize, in you need better compile speed and don't care about runtime speed use GetExtentSize.

/// w ##class(Utils.Persistent).GetGlobal("Utils.Persistent")
ClassMethod GetDataGlobal(Class As %Dictionary.CacheClassname) As %String
{
    Quit:'$$$comClassDefined(Class) ""
    Set Strategy = $$$comClassKeyGet(Class, $$$cCLASSstoragestrategy)
    Quit $$$defMemberKeyGet(Class, $$$cCLASSstorage, Strategy, $$$cSDEFdatalocation)
}

/// w ##class(Utils.Persistent).ExtentSizeFast("Utils.Persistent")
ClassMethod ExtentSizeFast(Class As %Dictionary.CacheClassname) As %String [ CodeMode = expression ]
{
$Get(@..GetDataGlobal(Class), 0)
}


/// w ##class(Utils.Persistent).GetExtentSize("Utils.Persistent")
ClassMethod GetExtentSize(Class As %String) As %Integer
{
    Set Global = ..GetDataGlobal(Class)
    Quit:Global="" 0
   
    Set Id = $Order(@Global@(""))
    Set Count = 0
    While Id '= "" {
        Set Id = $Order(@Global@(Id))
        Set Count = Count + 1
    }
    Quit Count
}

ClassMethod ExtentSize() As %Integer [ CodeMode = objectgenerator ]
{
    set Strategy = $$$comClassKeyGet(%classname, $$$cCLASSstoragestrategy)
    set Global = $$$defMemberKeyGet(%classname, $$$cCLASSstorage, Strategy, $$$cSDEFdatalocation)
    Do %code.WriteLine(" Set Id = $Order(" _ Global _ "(""""))")
    Do %code.WriteLine(" Set Count = 0")
    Do %code.WriteLine(" While Id '= """" {")
    Do %code.WriteLine("     Set Id = $Order(" _ Global _ "(Id))")
    Do %code.WriteLine("     Set Count = Count + 1")
    Do %code.WriteLine(" }")
    Do %code.WriteLine(" Quit Count")
    Quit $$$OK
}

In principle I'd share te suggestion of @Stephen Canzano;

with 2 minor additions: SELECT Count(ID) from <youtable>  will you always lead to the
explicit or implicit Extent Index.  (bitmap or standard)

This is important as there are a bunch of classes that have explicit defined table names.

In addition, it works also for table linked over SQL gateway