I don't fully understand the difference between factorizing a polynomial in irreducible factors in $\mathbb{Z}[X]$ and $\mathbb{Q}[X]$.
For example $f=X^4-X^2+4X+3$ is irreducible in $\mathbb{Z}[X]$, since $f\bmod2=X^4-X^2+1$ is irreducible in $\mathbb{F}_2[X]$. How would I prove this/factorize this polynomial in $\mathbb{Q}[X]$? Does irreducibility in $\mathbb{Z}[X]$ imply irreducibility in $\mathbb{Q}[X]$?
If a polynomial $f$ with integer coefficients has a factorization as product of two non-constant polynomials with rational coefficients, then it also has a factorization as a product of two non-constant polynomials with integer coefficients. This is basically Gauss's lemma mentioned in a comment already.
Thus if you can exclude that the polynomial factors into two non-constant integer polynomials you have shown it is irreducible over the rationals.
However, beware that "irreducible polynomial" in $\mathbb{Z}[X]$ is a subtle notion, as for example $7$ is an irreducible element of $\mathbb{Z}[X]$ that you may or may not want to consider as irreducible polynomial, or $2X-2$ is $2 \cdot (X-1)$ neither of which are units so $2X-2$ is not an irreducible element in $\mathbb{Z}[X]$.
Note the notion of "primitive polynomial" on the linked site to address these issues.