A large sheet with the words "Sack Hinkley" was plastered over the Power's roadside sign on Port Road near Alberton on Monday.
Port's finals hopes hang by a thread after Saturday's 12-point loss to Geelong left them in 11th spot with an 8-10 record.
Even if the Power win their remaining four games - against Collingwood, Richmond, Essendon, and Adelaide - they will still need other results to fall their way to sneak into the top eight.
Hinkley is contracted for next year, but the pressure is building on him following Port's yo-yo season.
Montgomery feels the heat will be "water off a duck's back" for Hinkley, and he expects the 10-year coach to still be at the helm in 2023.
"Yeah I would be shocked (if he's not here next year)," Montgomery said.
"He has a contract. It's a very good list, a good team.
"What we know about our supporter base is their passion levels are a little higher than most.
"Now, whether that's real or not, that's how we feel it, and that's a good thing.
"That passion, if that's boiled over to a sign on Port road, then that's just the passion our people have.
"He will push on. He's got belief in the group, and I think the group has belief in him."
Hinkley has been floated as a possible contender for the vacant North Melbourne role given his track record of being able to rebuild a team.
The 55-year-old led Port to preliminary finals in 2020 and 2021 after they finished 10th in 2016, 2018 and 2019.
"There's clubs out there looking for new coaches, and to look at a rebuild that's been as successful as this over 10 years, I think all clubs would be silly not to acknowledge what's been done here," Montgomery said of Hinkley being potentially head-hunted.
Port have been playing catch-up all year after starting their season with five straight losses.
They managed to square their season at 7-7, but have since lost three of their past four matches.
Montgomery says the team will try their best to stay in the finals hunt.
"I think the odds are definitely stacked against us, but that's been the case since our 0-5 start," Montgomery said.
"I don't expect anything to change in terms of our attitude or intensity.
"We do acknowledge it's going to be really tough and so many things need to fall our way."
Port will be back in action on Saturday facing the in-form Collingwood at the MCG.
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