Not all principal-agent problems are the result of incomplete contracts, no. In fact, in most principal-agent problems, complete contracts are assumed.
An incomplete contract is one that cannot be contingent on every possible outcome that could occur after the contract is signed. You can't write a contract that gives out a different payment to each party under every possible circumstance, because you can't imagine every possible circumstance. For example, an American agricultural firm's profits might be affected in miniscule ways by weather in Peru. But a contract related to that firm might offer the same payouts regardless of Peru's weather.
A principal-agent problem arises when the incentives of the principal and of the agent are not aligned, and so the agent may not act as the principal desires. For example, putting in less effort than the principal would find optimal because the agent will be the primary benefactor of additional effort. A lot of the problem arises here because the principal cannot perfectly and costlessly observe what the agent does, and so must write a contract contingent on the outcome of the project, rather than contingent on the agent's actions, which cannot be observed.
You could, colloquially, say that the principal-agent problem is an example of incomplete contracts in the sense that you can't contract on the agent's actions. But that's not what economists mean by incomplete contracts. Incomplete contracts refers to the fact that your contract is not sensitive to every possible contingency, even though everyone will know what that contingency is after it happens, as opposed to principal-agent problems, which refer to the fact that you can't include unobservable actions of the agent in your contract, which are hidden to the principal.
As for how they're related, they're both examples of problems that can make it difficult to write a contract. There are certainly models that combine the two (for example this), but one doesn't subsume the other.