In the comic strip Calvin and Hobbes, was there any moment where Calvin realised that Hobbes wasn't real?
The assumption in the question is mistaken.
As @Mithrandir already linked above, it's a common assumption of adult readers that Hobbes couldn't possibly be real but that's just not how the actual comic works. Watterson intentionally left the entire issue unresolvable, with strips like this
where Hobbes should need to be real for Calvin to be in the situation he's in. The shift between how Calvin and his parents see the world is very real and part of what the comic explores, but Watterson didn't feel the parents were right and Calvin was wrong.
What makes you think that he's not real?
First, consider what you mean by "real." If you mean "could you put a video camera there and record Hobbes doing the things that the comic strip has him doing with Calvin?", that's deliberately left ambiguous in the comic strips. In fact, even in a strip where Calvin takes photographs of Hobbes, Calvin and his dad just see the pictures differently, so it resolves nothing - it's not clear what the pictures "really" look like.
I'm not convinced that the "video-camera approach" is the best way to view this, though. Try to view it through the characters' POV. From Calvin's POV, yeah, he's real. From the other characters' POV, he's a stuffed tiger. I don't think that it's really important (or provable from the comic strip) which one is "correct" in the video camera sense.
As far as I understood when I read Calvin and Hobbes, the situations Calvin finds himself are always ambiguous. One could choose to believe that Calvin just has a high imagination in his role play with his stuffed toy or one could believe that Hobbes is some kind of magical toy that comes to life only for Calvin and is a stuffed toy to the rest of them. something like Perry in "Phineas and Ferb" (just the concept not the stuffed part).
And I always read these comics from the second point of view because it makes it that much more fun. Especially since we never hear dialogues from Hobbes when there are other people around. All his interactions are mostly done when they are alone.
I'd argue yes, in this strip (May 20, 1986):
Some answers are stating what the comic's author intended which is the characters never actually state this. Some may claim Susie does, but the conversation is between Calvin and her so from the reader's perspective it's a he said/ she said scenario and can't reliably answer the realness of Hobbes. Also, what an author's intent for their work doesn't matter. If what they write leads most readers to interpret the words differently, then they are horrible writers and couldn't convey the meaning they intended. What we can say to answer this question, at no time is it illustrated in the comics that Calvin views Hobbes as anything other than a "real" tiger. Anything else outside those panels is pure speculation including the author's intent.