I know that 'flirt' usually means, but I think here that word has a different meaning. I guess the sentences below seem to mean that the blue-jay tried to see where the acorn was disappearing. Here is the example, which is from a novel 'Baker's blue-jay yarn' by Mark Twain.
So he flew off and fetched another acorn and dropped it in, and tried to flirt his eye to the hole quick enough to see what become of it, but he was too late.