Are all subsets of NP-complete languages also NP-complete?
Take any infinite NP-complete language $L$. The language $L$ has uncountably many subsets, and particular there is some uncomputable subset $Q \subseteq L$. Since $Q$ is uncomputable, in particular it is not in NP, and so not NP-complete.

You can also try your example. Let $L$ consist of all pairs $(G,k)$ such that $G$ can be colored using at most $k$ colors, and let $Q$ consist of all pairs $(G,2)$ such that $G$ is bipartite; note that $Q \subseteq L$. It is known that $L$ is NP-complete, whereas $Q$ is in P. Therefore $Q$ is not NP-complete unless P=NP.