Is there a word for "bright colored eyes"?
Solution 1:
Use "Light-colored eyes"
As a native (American) English speaker, this would certainly convey "not brown or black". It may also exclude certain dark greens, which I'm not sure if the original Iranian phrase would.
Note that below some of the images are out of context, and appear to be used for comparison to dark eyes. I did not attempt to filter them out.
@tchrist has mentioned this in two comments.
Solution 2:
In the UK we don't have such a word. This is probably true in any English-speaking nation.
In Britain, eyes vary greatly in colour and so we don't consider there to be a standard or "colorless" eye hue. Some people have brown eyes, some people have blue eyes, some have hazel eyes, and so on.
I suggest the word "vivid" to describe eyes of a particularly bright hue.
For example, I would describe this person as having vivid green eyes.
This person has vivid blue eyes.
However the following person does not have vividly coloured eyes in my opinion. I would describe them as being pale-blue.