<p>This paper presents a significant extension of the classical Bedrosian identity to the quaternionic domain for functions of two variables. By leveraging the Quaternion Fourier transform, we develop a rigorous theoretical framework for the Quaternion Partial and Total Hilbert transforms. The core advantage of this Hilbert-based approach, as opposed to one using the rotational-invariant Riesz transform, is the simplicity of its Fourier multiplier. This property is fundamental and uniquely enables the derivation of Bedrosian-type identities, which are proven to be unattainable for the Riesz transform. We establish sufficient conditions for these identities to hold, providing a powerful multiplicative law for quaternionic signals under specific spectral conditions. Building upon this foundation, we delineate the necessary and sufficient conditions for the Quaternion Analytic Signal (QAS). Furthermore, as a key application of the Bedrosian theorems, we derive the precise criteria that ensure that the product of two holomorphic QASs remains a quaternion holomorphic function. The practical superiority of this framework is demonstrated through calculated examples and applications in two-dimensional image processing, where it offers a computationally effective and theoretically sound alternative to the monogenic signal, particularly for images with strong directional or lattice structures. This work provides essential theoretical tools for advancing hypercomplex signal processing and opens new avenues for sophisticated image analysis.</p>