Is there a problem solution which, if implementable in a given language, implies that that language is Turing complete?
The answer is no (although from a rather syntactic reason, which might not be what you were looking for).

Let $f : \Sigma^* \rightarrow \Sigma^*$ be some arbitrary function, then it is implementable in the following "stupid" language which is not Turing complete:

Any string $s\in \Sigma^*$ is a valid program, and all strings represent the program which, given $x$ as input, outputs $f(x)$. Note that I don't really care how it computes $f$ (as I'm defining a new model of computation). If this bothers you, imagine any string is translated to the same fixed Java program which computes $f$ (assuming $f$ is computable).