Yes. Tertiary/quaternary structure is a feature of both thermodynamics (hydrophobic portions of the protein folding in on themselves and hydrophilic portions of the protein exposing themselves to water, for example) and some clever folding tricks (e.g. making certain disulfide bonds) before being released to the cytoplasm (or wherever the protein is destined).
Beyond that, modifications such as phosphorylation and adenylation are cellular tricks to triggering further conformation changes of the proteins. Binding sites (both allosteric and the active sites) serve as another means to change conformation. Finally, interactions with other proteins, which is the important part, I believe, in prion type diseases, can also cause conformational changes.
If I had to guess, the interaction with a prion protein with a normal protein would cause the normal protein to shift conformation into the prion state and form a dimer of sorts which is more stable than the components are separated (or it's an irreversible change). Over time, this would result in the agglomeration seen with prions (as you form trimers, and on and on). Could be wrong though.
Edit: I should add - proteins are generally very flexible species. In solution, they're constantly contorting and twisting around and whatnot. Their "structure" is just either a) the structure of the crystallized protein (not flexible) or b) the average of the scans seen by NMR (in solution, so flexible).
