Is it possible to translate an object diagonally at a specific angle?

transform.Translate is mostly a shortcut to easily more your forward, or left or right. Once you start doing fancy stuff, it seems easier to change position directly.

To get a line going 30 degrees from right, for example:

 Vector3 A = Quaternion.Euler(0,30,0) * Vector3.right;

The first part assumes you have a top view. It rotates 30 degrees around Y. For a front view, use Z, the 3rd slot (or trial and error.)

Then, to move, just add:

 transform.position += A;

Can multiply A by speed, or whatever.

If you want to move 30 degrees from your current facing, the math is the same, but substitute "forward from me" for right:

 Vector3 A = Quaternion.Euler(0,30,0)*transform.forward;

You can definitely translate an angle in to a vector 3 using Mathf.sin or tan or cos whatever you need, it is just basic geometry.