As mentioned on the Wikipedia page for "integer triangle", the law of cosines forces the cosine of an angle in a triangle with integer sides to be a rational number.
Now, when does an integer angle (in degrees) have a rational cosine? We can go one step further: when does a rational angle (in degrees) have a rational cosine? This question is answered with a clever proof on page 2 of When is the (co)sine of a rational angle equal to a rational number? by Jörg Jahnel. The answer is that the rational cosines of the rational angles are just $\pm1,\pm\frac12,0$. (This is known as Niven's Theorem and an alternate proof can be found at ProofWiki.)
Since cosines of $\pm1$ wouldn't correspond to the angle of a genuine triangle, the only potential options are $\cos\theta=0,\pm\frac12$ for angles of $\theta=60^\circ,90^\circ,120^\circ$. But if there is one angle of $90^\circ$ or $120^\circ$, there isn't enough room in $180^\circ$ for two more angles of at least $60^\circ$. And the only triangles whose angles are all $60^\circ$ are equilateral.
The answer to the question "Can a non-equilateral triangle with integer sides and integer angles (in degrees) exist?" is no.