There is a position indicator that shows what position the flaps are in. If they are in something other than "up", then the green "LE Flaps Ext" light would also be illuminated.
Photo of 737 Flap Position Indicator
That's not to say that those indications might not be missed.
There are a couple of other things that would be not-quite-normal (i.e. VNAV information would be unusual, commanding a speed no faster than the speed limit for the current flap configuration), but if you're missing the one indication, you might well misinterpret the others.
That said, there is no aural warning for the flaps being oversped the way that there is for an MMO or VMO overspeed (the clacker) or an approach to stall (the stick shaker). The only warning per se would be on the speed tape on a PFD or EFIS display or in the HUD. And in a Classic (non-EFIS) 737 without the HUD in use, those wouldn't be seen.
The most obvious indications would be the leading edge flaps and leading edge slats rattling around, and the hit on performance during climb and cruise (the leading edge devices are a lot of drag). Generally that sort of thing is pretty hard to miss, but again, if you're having the kind of day where you're missing the call to retract the flaps and missing the position indicator & its green light, you might miss the others too.
My first reaction on reading that account was, wow that crew messed up big! On the other hand, the "quiet wing" mod mentioned in the other answer makes very good sense as an explanation -- especially if the OP does NOT remember the buffet from the leading edge devices banging around. Let's go with that one!
By 2005, I'd hope that nobody was pulling CB's in order to extend the trailing edge flaps only -- the cautionary tale of what happened to the 727 whose crew did that was pretty well known by then. (Essentially, somebody who wasn't "in" on the plan saw the CB out & pushed it in, extending the leading edge devices at high speed. One of them departed the aircraft, and the it took tens of thousands of feet, IIRC, to regain control of the aircraft. Bad scene!)