Not if you define a "biomolecule" as occurring naturally only in certain systems which we refer to as "biological systems" such as proteins. Obviously other molecules enter into biological processes, so if you want to call them "biomolecules" in this context, go ahead. The online definitions are not very discriminating, but they don't mention H2O in the examples I've seen. It's certainly true that water acts as a biomolecule.
Yes, water has such an active role in cellular processes that it is often considered a biomolecule. Although most biomolecules are carbon-based and water is not organic, it is naturally integrated into biological systems (natural occurrence).