No, it's definitely a John Noble thing, not a Walter Biship thing. It's also shown up in his narration of Dark Matters, a series completely unrelated to Fringe, and according to some places it was visible in works older than Fringe. _{(No reference here yet, as I have not seen the vast majority of what he's been in. I'll check the Stargate episode at some point, though)}

The running theory is indeed some sort of eye surgery, most likely LASIK, although I've not found any definitive quote.

IGN.com also has this to say on it:

Speaking of John Noble, I know at least one person is going to mention the oddness with his flickering eye in this episode. His eye has actually been like that for years. His odd ocular affliction is perhaps the result of some surgery, which was perhaps the result of a drunken fight with actor Omar Sharif, who reportedly smashed a table lamp in Noble's face during a meal after they wrapped up a day of filming on a movie they worked on together (further proof that John Noble is the most interesting man in the world). You can see Noble's trademark eye oddness in some Fringe episodes going all the way back to 2008 but it's particularly noticeable in a few scenes in this episode for some reason (lighting maybe?). Now, it remains to be seen whether the producers plan to incorporate this oddity into the story, but they've already featured that eye in previous episodes this season when Windmark tortured Walter and busted his eye. It could turn out to be something interesting for the story, but for now that glint in the old man's eye has a rather mundane explanation.

His eye had not been worked into the story by the end of the series.

No, it's definitely a John Noble thing, not a Walter Biship thing. It's also shown up in his narration of Dark Matters, a series completely unrelated to Fringe, and according to some places it was visible in works older than Fringe. _{(No reference here yet, as I have not seen the vast majority of what he's been in. I'll check the Stargate episode at some point, though) (Update 2017-05-25, see bottom of answer).}

The running theory is indeed some sort of eye surgery, most likely LASIK, although I've not found any definitive quote.

IGN.com also has this to say on it:

Speaking of John Noble, I know at least one person is going to mention the oddness with his flickering eye in this episode. His eye has actually been like that for years. His odd ocular affliction is perhaps the result of some surgery, which was perhaps the result of a drunken fight with actor Omar Sharif, who reportedly smashed a table lamp in Noble's face during a meal after they wrapped up a day of filming on a movie they worked on together (further proof that John Noble is the most interesting man in the world). You can see Noble's trademark eye oddness in some Fringe episodes going all the way back to 2008 but it's particularly noticeable in a few scenes in this episode for some reason (lighting maybe?). Now, it remains to be seen whether the producers plan to incorporate this oddity into the story, but they've already featured that eye in previous episodes this season when Windmark tortured Walter and busted his eye. It could turn out to be something interesting for the story, but for now that glint in the old man's eye has a rather mundane explanation.

His eye had not been worked into the story by the end of the series.

Stargate SG-1 9x20, Camelot aired 2006-03-10, about two and a half years before Fringe premiered. The flash was visible then:

And another shot about 4 seconds later, just to show the character didn't have a false eye or something: