1. Given that $P(x)$ and $Q(x)$ are non-constant polynomials with nonnegative coefficients, and the coefficients of $P(x)$ are not larger than 2021, while $Q(x)$ has at least one coefficient larger than 2021. We also know that $P(2022) = Q(2022)$ and both polynomials have a root $\frac{p}{q} \neq 0$ where $p, q \in \mathbb{Z}$ and $\gcd(p, q) = 1$.

2. Since $P(2022) = Q(2022)$, we can write:
$$Q(x) - P(x) = (x - 2022)(b_m x^m + b_{m-1} x^{m-1} + \cdots + b_0)$$
where $b_m, b_{m-1}, \ldots, b_0$ are coefficients.

3. The polynomial $Q(x) - P(x)$ is not constant because $Q(x)$ has at least one coefficient larger than 2021, and $P(x)$ has coefficients not larger than 2021. Therefore, $m > 0$.

4. The coefficients of $Q(x) - P(x)$ are:
$$b_m x^{m+1} + (-2022b_m + b_{m-1}) x^m + \cdots + (-2022b_1 + b_0) x + (-2022b_0)$$
Since the coefficients of $P(x)$ are not larger than 2021 and $Q(x)$ has at least one coefficient larger than 2021, all coefficients of $Q(x) - P(x)$ are $\geq -2021$.

5. We have the following inequalities:
$$b_m \geq -2021$$
$$-2022b_m + b_{m-1} \geq -2021$$
$$\vdots$$
$$-2022b_1 + b_0 \geq -2021$$
$$-2022b_0 \geq -2021$$

6. Assume $b_i \geq 1$ for some $i$. Then:
$$-2022b_i + b_{i-1} \geq -2021 \implies b_{i-1} \geq 2022b_i - 2021 \geq 2022 \cdot 1 - 2021 = 1$$
By continuing this process, we get $b_0 \geq 1$, which contradicts $-2022b_0 \geq -2021$. Therefore, $b_i \leq 0$ for all $i = 0, 1, \ldots, m$.

7. Thus, $b_m x^m + b_{m-1} x^{m-1} + \cdots + b_0 < 0$ for all $x > 0$. This implies $Q(n) - P(n) > 0$ for all $n = 1, 2, \ldots, 2021$.

8. Since $P(x)$ has nonnegative coefficients, its root $\frac{p}{q}$ must be negative. Assume $p > 0$, then $pn - q = |p| + n|q| > 0$ for all $n = 1, 2, \ldots, 2021$.

9. Given $P\left(\frac{p}{q}\right) = Q\left(\frac{p}{q}\right) = 0$, we can write:
$$Q(x) - P(x) = (px - q) H(x)$$
where $H(x)$ has integer coefficients.

10. We have:
$$(|p| + n|q|) H(n) = Q(n) - P(n) > 0 \text{ for all } n = 1, 2, \ldots, 2021$$
This implies $H(n) > 0$, so $H(n) \geq 1$ for all $n = 1, 2, \ldots, 2021$.

11. Therefore:
$$Q(n) - P(n) = (|p| + n|q|) H(n) \geq |p| + n|q| \text{ for all } n = 1, 2, \ldots, 2021$$

$\blacksquare$