I agree with Dennis.  I solved the equation for m and factored the resulting rational expression:

m = 4(n+1)(n-1)/n(3n-4)

Because m must be an integer, n(3n-4) has to divide evenly into 4(n+1)(n-1).  This means that all factors of n must divide BOTH the denominator AND the numerator here.  Because nonunit factors of n cannot also be factors of (n+1) or (n-1), we know that n must be a factor of 4.  Hence, I only have to check the cases n=1,-1,2,-2,4, and -4.  It is simple to show that the only possible value resulting in a nonzero integral value for m is n=2.

The solution is n=2, m=3.  I agree with Dennis' claim that it is unique.

Thank you K Sengupta for the puzzle!!

-John