No...not close to the trail.
As far as I know the upper Kern river is catch and release but down by lake Isabella you can catch trout and lots of other fish too.
I did not see any streams as we parked and hiked around the trees. But men can be blind.
There were no streams there, maybe you can find some if you go off trail though.
If you stick close to the trails along the road thwres no fishing spots. Maybe if you get into the forest a bit. But i didnt see anything like that. Honestly not much water at all. Its on the top of a mountain so I imagine all the water drains pretty quickly.
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