There is exactly one countable model (upto isomorphism) of the first order theory of $(\mathbb Q,<)$.
I am reading about spectrum of complete theories where the spectrum of various theories are stated (but not proved). I am looking for a reference where the above statement is proved, so that I can prove the other statements similarly, and so I can understand how can one prove such statements.
On
But that initial claim [before it was edited] isn't true. You know from the upward Lowenheim-Skolem theorem that any theory with a countable model also has models of all larger cardinalities (which therefore won't be isomorphic with the countable model).
What is true is that all countable models of the theory of dense linear orders without endpoints look the same. It is still false that there is exactly one model: there are still many models. But at least these, the countable models, are all isomorphic.
That special result is proved in any model theory text! Or googling we find e.g. https://www.math.ucsd.edu/~sbuss/CourseWeb/Math260_2012WS/Mar05Ricketts.pdf
The statement is proved in Theorem 2.5 here.