The main problem seems to be that JQA simply doesn't have enough opportunities. He was sixty-nine in 1836, so it was probably his last chance, but Jackson's popularity made Van Buren unbeatable. I guess the first thing is to mess up Jackson's second term, maybe have the Panic of 1837 start a couple of years earlier.
The anti-Jacksonians would be primed for victory, but JQA isn't popular enough in the West and North to unite the Whigs, let alone win outright as an Anti-Masonic candidate. Perhaps the strategy could be similar to OTL 1836: Western and Southern Whigs have their own candidates, but JQA becomes the Anti-Masonic nominee and gets the support of Northeastern Whigs (including New England and hopefully states like New York and Pennsylvania as opposed to just Massachusetts). The election goes to the House, but I'm not sure how he could get elected again there due to his aforementioned unpopularity in much of the country. Maybe it's van Buren vs. Harrison vs. Adams in the House, but Harrison's health problems start showing earlier (like Crawford in 1824). The Western Whigs decide that Harrison is a lost cause and Adams is the lesser evil to van Buren. But that's just a hypothetical; not sure how plausible it is.
As for the power of the Anti-Masonic Party as a whole, from what I understand they were formed partially out of partially legitimate concern for Free Mason practices (such as that their oaths would conflict with their public duties in office), but mostly out of paranoid panic promoted by opportunistic politicians. As a single-issue party, it was unsustainable and would inevitably merge with the Whigs or a similar force. However, the party itself could be strengthened in the short-term, which in turn could make Anti-Masonicism in general a more lasting force. Vermont elected an Anti-Masonic governor in 1831, as did Pennsylvania in 1835, so JQA's election as Massachusetts governor wouldn't be revolutionary, but it would certainly give the party a higher profile.