Is it possible to make template specialization for zero template arguments?

If T is used only for the constructor, you don't need to template the whole class:

#include <iostream>

struct S {
  int n = 1;

  template <typename T>
  S(T t) : n(t) {};

  S() = default;
};

int main() {
  S s1 {10};
  std::cout << "Value:\n" << s1.n << std::endl;
  S s2 {};
  std::cout << "Value:\n" << s2.n << std::endl;
}

You might specialize S for void and create a CTAD https://en.cppreference.com/w/cpp/language/class_template_argument_deduction

#include <iostream>

template <typename T>
struct S {
  int n = 1;
  S(T t) : n(t) {}; // no default
};

template <>
struct S<void> {
  int n = 1;
  S() = default;  // only default
};

// CTAD calls to constructor S() will instantiate as S<void> 
template<typename... T> S() -> S<void>;   

int main() {
  S<int> s1 {10};
  std::cout << "Value:\n" << s1.n << std::endl;
  S s2 {};  // here CTAD will be trigged
  std::cout << "Value:\n" << s2.n << std::endl;
}

A link to cppinsights may help understand what and where things are being instantiated: https://cppinsights.io/s/8f0f4bf6