Let $I$ be an ideal of $\mathbb{Q}[T_1,\dotsc,T_n]$, and let $J=I\cap\mathbb{Z}[T_1,\dotsc,T_n]$.
Consider $X=\operatorname{Spec}(\mathbb{Q}[T_1,\dotsc,T_n]/I)$ and $Y=\operatorname{Spec}(\mathbb{Z}[T_1,\dotsc,T_n]/J)$.
Then $Y\times_{\operatorname{Spec}(\mathbb{Z})}\operatorname{Spec}(\mathbb{Q})=X$.
In this way an affine $\mathbb{Q}$-scheme $X$ gives rise to an affine $\mathbb{Z}$-scheme $Y$ such that $Y\times_{\operatorname{Spec}(\mathbb{Z})}\operatorname{Spec}(\mathbb{Q})=X$.
I am wondering if this construction of $Y$ from $X$ has a name in scheme theory.
(I am an absolute beginner)
First of all, this is not a well-defined construction on a $\mathbb Q$-scheme $X$. That is, the construction depends on the presentation of the scheme. For example, $\mathrm{Spec}\ \mathbb Q[x]/x$ and $\mathrm{Spec}\ \mathbb Q[x]/(2x-1)$ are clearly isomorphic, but the construction gives $\mathrm{Spec}\ \mathbb Z[x]/x\cong\mathrm{Spec}\ \mathbb{Z}$ and $\mathrm{Spec}\ \mathbb Z[x]/(2x-1)\cong\mathrm{Spec}\ \mathbb Z_{(2)}$, respectively, which are not isomorphic.
However, such problems are important in scheme theory, namely: given a morphism $S'\to S$ and a scheme $X'$ over $S'$, when does $X'$ arise as a fiber product $X\times_SS'$ for a scheme $X\to S$, and if so, can you explicitly construct such a scheme $X$?
Examples are when $S'=\mathrm{Spec}(L)$ and $S=\mathrm{Spec}(K)$, where $L/K$ is a Galois extension. Then the problem is usually known as Galois descent, and is analogous to the theory of Galois descent for vector spaces.
When $S'=\mathrm{Spec}(k)$ and $S=\mathrm{Spec}(k[x]/x^2)$ (or perhaps the formal scheme $\mathrm{Spf} k[[x]]$) for some field $k$, then the problem is usually called deformation theory.
In your particular example, we have $S'=\mathrm{Spec}\ \mathbb Q$ and $S=\mathrm{Spec}\ \mathbb Z$. You give one construction for affine schemes over $S'$, but I believe extending the construction to arbitrary $\mathbb Q$-schemes is still not easy, since it is unclear how the construction on the affine patches glue together (precisely because the construction is on the level of presentations, not on the level of rings). Here, I believe the key word is integral models.
Here is a geometric interpretation of OP's construction. Let $X\hookrightarrow\mathbb A_{\mathbb Q}^n$ be a closed subscheme. There is a morphism $\mathbb A_{\mathbb Q}^n\subset\mathbb A_{\mathbb Z}^n$. The composition is a morphism $X\to\mathbb A_{\mathbb Z}^n$, and the scheme-theoretic image is your $Y$ (see the Stacks project.)
This construction generalizes: Let $Z\hookrightarrow X$ be a closed immersion of $\mathbb Q$-schemes. If $X=X'\times_{\mathrm{Spec}\ \mathbb Z}\mathrm{Spec}\ \mathbb Q$ for some $\mathbb Z$-scheme $X'$, then $Z'$, the scheme-theoretic image of $Z\hookrightarrow X\to X'$ is such that $Z=Z'\times_{\mathrm{Spec}\ \mathbb Z}\mathrm{Spec}\ \mathbb Q$.