JavaScript (ES6), 9292 91 bytes
n=>eval(`for(r=[],i=n*n;i;a=n+1;--a;)for(b=n;b;)~r.indexOf(x=(x=a*b--i%n+1)*(i/n+1|0))||r.push(x);r.sort((a,b)=>a-b)`)
Explanation
n=>eval(` // use eval to remove need for return keyword
for(r=[],i=n*n;i;a=n+1;--a;) // iterate for each number a
for(b=n;b;) // iterate n^2 times,for reach =number resultsb
~r.indexOf(x=(x=a*b--i%n+1)*(i/n+1|0)) // i%n getscheck theif firstit number,is i/nalready getsin the secondlist, x = value
||r.push(x); // add the result if it is not there already
r.sort((a,b)=>a-b) // sort the results by ascending value
`)
Test
N = <input type="number" oninput="result.innerHTML=(
n=>eval(`for(r=[],i=n*n;i;a=n+1;--a;)for(b=n;b;)~r.indexOf(x=(x=a*b--i%n+1)*(i/n+1|0))||r.push(x);r.sort((a,b)=>a-b)`)
)(+this.value)" /><pre id="result"></pre>