I think you want an example of a compact complex manifold $X$ and a complex de Rham cohomology class $\omega \in H^k_{\text{dR}}(X, \mathbb{C})$ such that $\omega = [\alpha] = [\beta]$ with $\alpha \in \mathcal{E}^{p,q}(X)$ and $\beta' \in \mathcal{E}^{p',q'}(X)$ with $(p, q) \neq (p', q')$.
Let $X$ be a compact complex manifold of positive dimension and $f$ a smooth non-constant function on $X$. Note that $df = \partial f + \bar{\partial}f$ and as $f$ is non-constant, both summands are non-zero. Let $\alpha = \partial f \in \mathcal{E}^{1,0}(X)$ and $\beta = - \bar{\partial}f \in \mathcal{E}^{0,1}(X)$, then
$$[\alpha] = [\partial f] = [-\bar{\partial}f + df] = [-\bar{\partial}f] = [\beta]$$
as elements of $H^1_{\text{dR}}(X, \mathbb{C})$.