P is undecidable and not semidecidable, Q is undecidable and semidecidable and P ⊂ Q
Hint: Following the OP's suggestion, let $P$ be the set of all TMs halting on every input, and let $Q$ be the set of all TMs halting on the empty input. It is known that $Q$ is undecidable and semidecidable (indeed, $\Sigma_1^P$-complete) and that $P$ is undecidable and not semidecidable (indeed, $\Pi_2^P$-complete). Moreover, $P \subseteq Q$ since if $T \in P$ then $T$ halts on every input, and in particular $T$ halts on the empty input, so that $T \in Q$.