  We extend the adiabatic regularization method for an expanding universe to
include the Yukawa interaction between quantized Dirac fermions and a
homogeneous background scalar field. We give explicit expressions for the
renormalized expectation values of the stress-energy tensor $\langle T_{\mu\nu}
\rangle$ and the bilinear $\langle \bar\psi\psi\rangle$ in a spatially flat
FLRW spacetime. These are basic ingredients in the semiclassical field
equations of fermionic matter in curved spacetime interacting with a background
scalar field. The ultraviolet subtracting terms of the adiabatic regularization
can be naturally interpreted as coming from appropriate counterterms of the
background fields. We fix the required covariant counterterms. To test our
approach we determine the contribution of the Yukawa interaction to the
conformal anomaly in the massless limit and show its consistency with the heat
kernel method using the effective action.
