All decision problems (i.e.language membership problems), which are verifiable in polynomial time by a deterministic Turing machine are called NP problems. Further, these problems can be solved by a non-deterministic Turing machine in a polynomial time and in exponential time by a deterministic Turing machine.
Do we have a decision problem that is not verifiable by a deterministic Turing machine in polynomial time but decidable?
No to the question in your title; yes to the question in the body of your post.
By the non-deterministic time hierarchy theorem, every decidable NEXP-hard decision problem is "a decision problem that is not verifiable by a deterministic Turing machine in polynomial time but decidable".
A concrete counterexample could be Quantifed Boolean Formula, which is only verifiable in polynomial time if PSPACE=P. There's a whole myriad of other examples. You might also want to look at coNP-complete problems. For these a polynomial TM can only verify counterexamples (unless coNP=NP). The language of all tautologies is a prominent example for that class.
