Why does my while loop never end, even while the produced integer fails the conditions to run

Your loop has no exit condition because you don't test the return value of std::string::find properly.

If you read the manual, you'll see that the return value when no match is found is std::string::npos.
Don't expect it to have any other value than the constant provided by the STL, or you'll be sorry.

This will work:

#include <iostream>
#include <string>    // include what you need instead of relying on other includes
//using namespace std; <--- don't do this unless you know exactly what you're doing
//                          (which is unlikely given your limited acquantaince with C++)
int main(void) // "void" is optional, but indicates (argc,argv) have been purposely ignored
{
    std::string phrase = "a Hello I am a pineapple a"; // test edge cases when possible
                                                       // (here, the first and last chars)
    size_t position = 0;  // "num" is a terrible variable name, like "int someint"
                          // and the proper type to use is size_t (unsigned).
                          // Using an int will get you a warning.
                          // Ignore warnings at your own risk...
    while (true)
    {
        position = phrase.find('a', position);    // look for the character
        if (position == std::string::npos) break; // exit condition
        std::cout << position << std::endl;       // display found occurrences
        position++; // move to next character (or end of string)
    }
}

Or if for loops are more your thing, you can slightly reduce the code like so:

for (size_t position = 0 ; ; position++)
{
    position = phrase.find('a', position);
    if (position == std::string::npos) break;
    std::cout << position << std::endl;
}

Some people find that nicer, some harder to read.
I rather like it because it makes position local to the loop, preventing you from doing something unwise like reusing the variable outside the loop.
It's really a matter of preference.

The issue was that (like @kalyanswaroop said) when find() cannot find a character it returns -1, the next step in the loop changes the assigned value from -1 to 0, and so it satisfies the conditions of the loop. In order to prevent this from happening I changed the code to the below, so that if num is -1, it breaks the loop.

#include <iostream>
using namespace std;

int main() {
    string phrase = "Hello I am a pineapple";
    
    int phraseLength = int(phrase.length() - 1);
    int num = 0;
    
    
    while(num >= 0)
    {
        
        num = int(phrase.find(char('a'), num));
        if(num > -1)
        {
            cout << num << endl;
            num++;
        }
        else{
            break;
        }

    }
  
    return 0;

}

Suggestion: Change your code

Here is a few suggestions

Using find()

The by far easiest way to do this is by using Find() (Thanks to neko for suggesting this. Not sure if he really suggested that)

Example:

#include <iostream>
#include <string>
#include <algorithm> //Don't know if this is required
int main()
{
     string phrase = "Hello I am a pineapple";
     auto location = std::find(phrase.begin(), phrase.end(), 'a');
     cout << *location;
}

Creating a function like find()

Another way to do this (also to get a taste on how find() might be implemented) is by writting a function similar to find()

Example:

#include <iostream>
#include <string>
using namespace std;

template<typename Iterator, typename T>
Iterator Find_Character(Iterator begin, Iterator end, T Find)
{
   while (begin != end && *begin != Find) ++begin;
   return begin;
}

int main()
{
    string phrase = "Hello I am a pineapple";
    auto location = Find_Character(phrase.begin(), phrase.end(), 'a');
    std::cout << "Found character: " << *location << endl; 
}

when the phrase.find cant find 'a', it returns -1, and then you have num++ which brings num back to the same condition as the beginning. Perhaps you should do something like:

while(num < phrase.length() -1 && num > -1)
{
    int findPos = int(phrase.find('a', num));
    if(findPos != nPos) //nPos is defined as -1
    {
            num = findPos ;
    }
    else
    {
            break ;
    }
    cout << num << endl;
    num++;

}