Complex conjugate

Is this what you wanted?

expr = E^(-((I β)/2)) p (Cos[α/2] (Cos[θ]^2 + Sin[θ]^2 Sin[ϕ] (-I Cos[ϕ] + Sin[ϕ])) + 
    E^(I β) Sin[α/2] (Cos[θ]^2 + Sin[θ]^2 Sin[ϕ] (I Cos[ϕ] + Sin[ϕ])))

expr /. Complex[x_, y_] :> Complex[x, -y]

You might use:

  FullSimplify@ComplexExpand@Conjugate[(* expression *)]

in your case, it returns:

$$ e^{\frac{i \beta }{2}} p \left(\sin \left(\frac{\alpha }{2}\right) e^{-i (\beta -\phi )} \left(\cos ^2(\theta ) \cos (\phi )-i \sin (\phi )\right)+\cos \left(\frac{\alpha }{2}\right) \left(\cos ^2(\theta )+\sin ^2(\theta ) \sin (\phi ) (\sin (\phi )+i \cos (\phi ))\right)\right) $$

You can also do

Refine[Conjugate@c[[1]],
       Assumptions->(\[Alpha] | \[Beta] | \[Phi]) \[Element] Reals]

This also works in cases when you have additional manifestly complex variables in your expression (ComplexExpand assumes that all variables are real). Of course in that case you would not add those in your Assumptions. For example:

Refine[Conjugate[a b], Assumptions -> a \[Element] Reals]
(*
==> a Conjugate[b]
*)