(Using Barbecue 1.5beta1 on Windows XP)
net.sourceforge.barbecue.linear.code39.Code39Barcode does not generate quiet sections, even if the drawingQuietSection field is set to true. I noticed that the getPreAmble() and getPostAmble() methods don't implement any quiet section logic, unlike, for example, Code128Barcode.
I'm new to barcodes so don't know if the omission of quiet space from Code39BarCode is deliberate or not, but I've included a subclassed version of Code39Barcode which does implement quiet sections by overriding getPreAmble() and getPostAmble(), and I'd be happy to convert it to a patch if you'd find it useful.
FixedCode39Barcode.java