Is there any difference between the sentences below?
Sie denkt, sie könne schwimmen.
Sie denkt, sie konnte schwimmen.
If there is, what is it?
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Sie denkt, sie könne schwimmen.
This means the considers herself able to swim.
Note that these days often people say instead:
Sie denkt, sie kann schwimmen.
I'm not sure whether this is also formally allowed, though.
Sie denkt, sie konnte schwimmen.
This means she thinks she was able to swim at some time in the past.
Note that there is yet another form:
Sie denkt, sie könnte schwimmen.
This means she falsely considers herself able to swim.
If you use these phrases in spoken German, there is no large difference. But there still is one. The first sentence means:
The girl thinks that she is able to swim.
The second sentence does emphasize that she considers herself to be able to swim in the past.
Grammatically, this sentence is a perfect example for when to use the Konjunktiv 1, because it is indirect speech.
Das Mädchen denkt: "Ich kann schwimmen".
Das Mädchen denkt, sie könne schwimmen.
Thus, the first sentence is correct.