An exponential Lévy process is typically modelled via $$ S_t = S_0\exp\left(\left(r-q+\omega\right)t+X_t\right),$$ where $X_t$ is a Lévy process with $X_0=0$. A Lévy process includes three model features: a linear drift, diffusive shocks and jumps (which may be large and rare or small and frequent). The number $\omega$ is called martingale correction or Jensen's correction and ensures the martingale property.
For our standard finance theory to work, the reinvested and discounted stock price, $S_te^{-(r-q)t}$, needs to be a martingale under $\mathbb{Q}$ (assuming constant interest rates and dividend yields). Let $(\mathcal{F}_t)$ denote the natural filtration of $X_t$. Then, for any $s\leq t$,
\begin{align*}
\mathbb{E}^\mathbb{Q}[S_t|\mathcal{F}_s] &= \mathbb{E}^\mathbb{Q}[S_0e^{(r-q+\omega)t+X_s+(X_t-X_s)}|\mathcal{F}_s] \\
&= S_0e^{(r-q+\omega)t} e^{X_s} \mathbb{E}^\mathbb{Q}[e^{X_t-X_s}] \\
&= S_s e^{(r-q+\omega)(t-s)} \mathbb{E}^\mathbb{Q}[e^{X_{t-s}}],
\end{align*}
where we used that $X_s$ is $\mathcal{F}_s$-measurable, and $X_t-X_s\overset{d}{=} X_{t-s}$ is independent of $\mathcal{F}_s$, see here.
Let $\varphi_{X_t}(u)=\mathbb{E}[e^{iuX_t}]$ be the characteristic function of the Lévy process $X_t$. The Lévy-Khintchine formula states that $\varphi_{X_t}(u)=e^{t\Psi(u)}$ which follows from the infinite divisibility of a Lévy process. The function $\Psi$ is called the characteristic exponent and captures the drift, diffusion and jump components of $X_t$.
Then,
\begin{align*}
\mathbb{E}^\mathbb{Q}[S_t|\mathcal{F}_s] &= S_s e^{(r-q+\omega)(t-s)} \varphi_{X_{t-s}}(-i) \\
&= S_s e^{(r-q+\omega)(t-s)} e^{(t-s)\Psi(-i)}.
\end{align*}
Hence, setting $\omega=-\Psi(-i)$ yields
\begin{align*}
\mathbb{E}^\mathbb{Q}[S_t|\mathcal{F}_s] &= S_s e^{(r-q)(t-s)},
\end{align*}
which in turn implies that the discounted reinvested stock price is indeed a $\mathbb{Q}$-martingale.
Note that $$\omega=-\Psi(-i)=-\frac{1}{t}\ln\left(\varphi_{X_t}(-i)\right)$$ is independent of time. Thus, for an exponential Lévy process, the martingale property is ensured to hold if you verify that $\mathbb{E}^\mathbb{Q}[S_t]=S_0e^{(r-q)t}$.
