1. The rationals are countable
2. The set of open intervals with rational end points is equivalent in the obvious way to a subset of the set of all ordered pairs of rationals.
3. Because of 1. there is a 1-1 onto map from NxN to QxQ.
4. NxN is countable hence there is a 1-1 onto map from N to NxN
5. from 4. and 3. there is a 1-1 onto map from N to QxQ, hence QxQ is countable.
6. A subset of a countable set is countable hence; the set of open intervals with rational end points is countable.
(proof of 4. is an easy modification of the usual proof of the countability of Q).
(note the "1-1" wherever it appears above is redundant "onto" will do, but its more intuitive if we leave the "1-1" conditions in)