Yes, my solution was more or less the same.
As for the last bit - equations of the form
n^2 - d*m^2 = 1, where d is not a square of an integer,
are the simplest form of something called pellian equations. It's not difficult to solve them, but it probably requires a bit of specialist knowledge. Brief explanation with a simple example can be found in this Wikipedia entry:
The basic idea is to find the least positive nontrivial solution (ie (n, m) is not (1, 0)), called the fundamental solution. Formally this can be done by finding the continued fraction representation of sqrt(d) and checking the successive convergents of sqrt(d). But in our particular case it is easy to "guess" one: (n, m) = (7, 2). Now the following theorem can be applied:
If (r, s) is the fundamental solution of n^2 - d*m^2 = 1, where d is positive and nonsquare, then every solution to n^2 - d*m^2 = 1 is given by (n(k), m(k)) where
n(k) +m(k)*sqrt(d) = (r + s*sqrt(d))^k for k = 1, 2, 3, ...
That is, raising [7 + 2*sqrt(12)] to the power of 2, 3, 4, ... will give all further integral solutions of n^2 - 12*m^2 = 1. And there is only one with n between 100 and 10000, namely (n, m) = (1351, 390).
Alternatively, of course, one can simply run through several thousand of m values and check if any of corresponding n values is an integer in (100, 10000) using some computer program, just as you did. It may be even easier in this particular exercise!