= Solution
Set $A=kQ$, $X_i=e_iX$, and let $f_\rho$ be the action of the arrow $\rho$ on $X_{s(\rho)}$. Write $P_i=Ae_i$, a left <projective module>. The <standard projective resolution of a quiver representation> is
$$
\boxed{0\longrightarrow\bigoplus_{\rho\in Q_1}P_{t(\rho)}\otimes_kX_{s(\rho)}\xrightarrow{d}\bigoplus_{i\in Q_0}P_i\otimes_kX_i\xrightarrow{\varepsilon}X\longrightarrow0.}
$$
The algebra acts on the first tensor factor. The augmentation is $\varepsilon(p\otimes x)=px$. On the summand for $\rho:i\to j$, the differential sends $p\otimes x$ to $p\rho\otimes x$ in the $i$-summand minus $p\otimes f_\rho(x)$ in the $j$-summand. Here $p\in Ae_j$, so $p\rho\in Ae_i$. These formulas specify every term and map, and $\varepsilon d=0$.
Exactness is the standard path resolution fact: the relations identify a path acting on a vector with successively applying its arrows; uniqueness of the first traversed arrow supplies injectivity of the relation map. Each $Ae_i$ is a direct summand of $A$ because $e_i$ is an <idempotent>, and tensoring with a $k$-vector space gives a direct sum of copies of $Ae_i$. Both terms preceding $X$ are consequently <projective modules>. Thus the displayed sequence is a <projective resolution> of length at most one.
The same path resolution works for arbitrary left modules, with possibly infinite-dimensional $X_i$ and arbitrary direct sums of <projective modules>. Hence every left $A$-module has projective dimension at most one. Equivalently, \b[$A$ is a left hereditary ring]: if $N\subseteq P$ with $P$ projective, dimension shifting gives $\operatorname{Ext}^1_A(N,M)\cong\operatorname{Ext}^2_A(P/N,M)=0$ for every $M$, so $N$ is projective. This argument also covers <quivers> with oriented cycles.
For dimension vectors $\mathbf n,\mathbf m\in\mathbb R^{Q_0}$, define the <Ringel form>
$$
\boxed{\langle\mathbf n,\mathbf m\rangle_Q=\sum_{i\in Q_0}n_im_i-\sum_{\rho\in Q_1}n_{s(\rho)}m_{t(\rho)}.}
$$
Apply $\operatorname{Hom}_A(-,Y)$ to the <projective resolution>. Evaluation at $e_i$ identifies $\operatorname{Hom}_A(P_i\otimes_kV,Y)$ with $\operatorname{Hom}_k(V,Y_i)$. Consequently there is an exact sequence
$$
0\longrightarrow\operatorname{Hom}_Q(X,Y)\longrightarrow C^0\xrightarrow{\delta}C^1\longrightarrow\operatorname{Ext}^1_Q(X,Y)\longrightarrow0,
$$
where
$$
C^0=\bigoplus_i\operatorname{Hom}_k(X_i,Y_i),\quad C^1=\bigoplus_\rho\operatorname{Hom}_k(X_{s(\rho)},Y_{t(\rho)}),\quad(\delta h)_\rho=g_\rho h_{s(\rho)}-h_{t(\rho)}f_\rho.
$$
This is the <extension complex of quiver representations>. Taking its alternating dimension sum yields
$$
\boxed{\dim\operatorname{Hom}_Q(X,Y)-\dim\operatorname{Ext}^1_Q(X,Y)=\langle\dim X,\dim Y\rangle_Q.}
$$
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