There is a certain ambiguity in your question. You see that the implicit actions that the compiler uses with respect to constructors include both declaring them and defining them. If any constructor is declared but not defined, do you consider it existing or not?
In any case, there is no way to create a class for which constructors are not declared. A copy constructor, for example, is always declared. There is no way to crush him. If you do not declare it yourself, the compiler will declare it for you.
As for the default constructor, you can disable its implicit declaration. If you declare any constructor yourself (i.e. Explicitly), the compiler will not declare a default by default. But in this case, your class, of course, will have a constructor: one that you yourself declared. (Also, as I said above, the copy constructor is always declared).
As for implicitly defined constructors ... They are defined by the compiler only if you use them. And, of course, they are determined only if this is possible. (If you use an implicit constructor, and it is impossible to determine, then your program will simply not compile).
So again, when it comes to declared constructors, it is impossible to have a class without any constructors. Any class has at least one constructor declared for it.
If you are interested in specific constructors specifically, then it is really possible to have a class for which a constructor is not defined. Here is an example
struct S { S(const S&); };
What is it. The class contains one constructor, but it is not defined :)