The interval [0,1) for example is uncountable, but we are not counting ELEMENTS of an interval, we are counting INTERVALS. [0,1] is one interval, [0,1/3] is another and so on.

Start with one end-point

Just count the left end points of [a,b].  Now for a fixed a there are countably many b’s which will form [a,b] as Q is countable. Now vary a∈Q. So there are countable union of countable elements.

Summary

There are a countable number of intervals (in R) with endpoints in the Rationals (Q).

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