Is the length of the shortest quine in a programming language computable?
You need to formally define the computational problem. It appears that the input is "a programming language" and the output is the length of the shortest quine in that language. But how is the programming language specified? The only way I can think of is to specify it by giving the encoding of a Turing machine that interprets the language. So, then, our input is $\langle M\rangle$ and our output is $\min\,\{|x|\mid M(x)=x\}$. This is uncomputable: it's uncomputable even whether $M(0)=0$