Yes. Make the function templates and then conditionaly enable them using std::enable_if:
#include <type_traits>
template <class T> 
class A {
public:
  template<typename U = T>
  typename std::enable_if<std::is_same<U,int>::value>::type
  has_int() {}
  template<typename U = T>
  typename std::enable_if<std::is_same<U,char>::value>::type
  has_char() {}
};
int main()
{
    A<int> a;
    a.has_int();   // OK
    // a.has_char();  // error
}
The solution from the other answer might not be feasible if the class is big and has got many functions that need to regardless of T. But you can solve this by inheriting from another class that is used only for these special methods. Then, you can specialize that base class only.
In C++14, there are convenient type aliases so the syntax can become:
std::enable_if_t<std::is_same<U, int>::value>
And C++17, even shorter:
std::enable_if_t<std::is_same_v<U, int>>