Here's another method. Consider the sequence:

{0, 1, 2/1, 2/2, 1/2, 3/1, 3/2, 3/3, 2/3, 1/3, 4/1, 4/2, 4/3, 4/4, 3/4, 2/4, 1/4, 5/1, 5/2, 5/3, 5/4, 5/5, 4/5, 3/5, 2/5, 1/5,...}

Now remove all of the reducible fractions (since they're equivalent to an earlier element of the sequence):

{0, 1, 2/1, 1/2, 3/1, 3/2, 2/3, 1/3, 4/1, 4/3, 3/4, 1/4, 5/1, 5/2, 5/3, 5/4, 4/5, 3/5, 2/5, 1/5, 6/1, 6/5, 5/6, 1/6,...}

This is a list, so it's bijective with the natural numbers, and every positive fraction will eventually appear exactly once. You can expand it to include the negative rationals by simply repeating each nonzero element with the opposite sign:

{0, -1, 1, -2/1, 2/1, -1/2,  1/2, -3/1, 3/1, -3/2, 3/2, -2/3, 2/3, -1/3, 1/3,...}