Initializing an array in a object in C++

The constructor you have implemented expects an int* pointer as input, but {2, 3, 4} does not decay to an int*, so no, the syntax you have shown will not work given the class you have implemented so far.

If you know the exact array size at compile-time, you could do this using std::array instead:

#include <array>

template<typename T, size_t N>
struct ary {
    std::array<T, N> data;
    ...
};

#include "ary.h"

int main()
{
    ary<int, 3> array_int1{2, 3, 4};

    ary<int, 3> array_int2 = {2, 3, 4};

    ary<int, 3> array_int3 = {{2, 3, 4}};

    return 0;
}

Otherwise, if you really want ary to have a pointer to some array data, you could declare an actual array first, and then pass it in to the constructor, eg:

template<typename T>
class ary {
private:
    T *dataptr;
    ...

public:
    ary(T* ptr) : dataptr(ptr) {}
    ...
};

#include "ary.h"

int main()
{
    int data[] = {2, 3, 4};
    ary<int> array_int(data);
    ...

    return 0;
};

But consider giving your class a constructor that takes a std::initializer_list as input instead, and then have the class allocate its own array internally (just be sure to follow the Rule of 3/5/0), eg:

#include <algorithm>
#include <utility>

template<typename T>
class ary {
private:
    T *dataptr = nullptr;
    int datasize = 0;

public:
    ary() = default;

    ary(const ary &src)
        : ary()
    {
        if (src.dataptr && src.datasize > 0)
        {
            dataptr = new T[size];
            datasize = src.datasize;
            std::copy(src.dataptr, src.dataptr+src.datasize, dataptr);
        }
    }

    ary(ary &&src)
        : dataptr(src.dataptr), datasize(src.datasize)
    {
        src.dataptr = nullptr;
        src.datasize = 0;
    }

    ary(T* ptr, int size)
        : dataptr(new T[size]), datasize(size)
    {
        std::copy(ptr, ptr+size, dataptr);
    }

    ary(std::initializer_list<T> l)
        : dataptr(new T[l.size()]), datasize(l.size())
    {
        std::copy(l.begin(), l.end(), dataptr);
    }

    ~ary()
    {
        delete[] dataptr;
    }

    ary& operator=(ary rhs)
    {
        std::swap(dataptr, rhs.dataptr);
        std::swap(datasize, rhs.datasize);
        return *this;
    }

    ...
};

#include "ary.h"

int main()
{
    ary<int> array_int1;

    ary<int> array_int2 = {2, 3, 4};

    int data[] = {2, 3, 4};
    ary<int> array_int3{data, 3};

    ary<int> array_int4{array_int2};

    ary<int> array_int5{std::move(array_int3)};

    ...

    return 0;
}

A better option is to use std::vector instead, and let it do all the work for you, eg:

#include <vector>

template<typename T>
class ary {
private:
    std::vector<T> data;

public:
    ary() = default;

    // the copy/move constructors, copy/move assignment operators,
    // and destructor will be implicitly generated for you...

    ary(T* ptr, int size) : data(ptr, size) {}
    ary(std::initializer_list<T> l) : data(l) {}

    ...
};

#include "ary.h"

int main()
{
    ary<int> array_int1;

    ary<int> array_int2 = {2, 3, 4};

    int data[] = {2, 3, 4};
    ary<int> array_int3{data, 3};

    ary<int> array_int4{array_int2};

    ary<int> array_int5{std::move(array_int3)};

    ...

    return 0;
}