Template constructor in a class template – how to explicitly specify template argument for the 2nd parameter?

Fixing your code, the following would work:

template<class T> class TestTemplate {
public:
    //constructor 1
    template<class Y> TestTemplate(Y * p) {
        cout << "c1" << endl;
    }

    //constructor 2
    template<class Y, class D> TestTemplate(Y * p, D d) {
        cout << "c2" << endl;
    }

    template<class A, class B>
    void foo(A a, B b) {
        cout << "foo" << endl;
    }
};

int main() {
    TestTemplate<int> tp(new int());

    tp.foo<int*, string>(new int(), "hello");

    TestTemplate<int> tp2(new int(),2);
}

You cannot use T for the class template parameter and the constructor template parameter. But, to answer your question, from [14.5.2p5]:

Because the explicit template argument list follows the function template name, and because conversion member function templates and constructor member function templates are called without using a function name, there is no way to provide an explicit template argument list for these function templates.

Therefore, you cannot explicitly specify template arguments for constructor.