Use the usual formula to solve as a quadratic in x.
You need the term under the square root sign to be a perfect square.
Wolfram Alpha gives a complete set of integer solutions. They look quite messy and I haven't checked them out in any way.
The first few though are not difficult following the quadratic/discriminant route.
You need to be a square, so check out the sequence .
gets you an early hit and a second comes not much further on.
How did you code your question to Wolfram|Alpha? I tried to do that to check my work but was unable to get it to solve the problem
I'm not a regular user of Wolfram Alpha, in fact this is the first time that I've actually asked it a question.
I simply typed in the equation and pressed the compute button.
It came back with a whole pile of stuff, including a graph, which was pleasing to see, and a section headed 'Integer Solutions '.