I'm thinking about mounting my surge suppressor inside a closed cabinet with very little air flow/circulation. Will this be a problem for my various power bricks that are connected?
That depends on 2 things.
1.) General wiring and power issues, e.g. do you have 1000+ watts of power going through the surge suppressor.? Remember that electrical wiring itself heats up via resistance that is driven by high power draw, so that could create heat, if its just computer equipment that shouldn't pose a problem, but if you have a fridge/freezer and an air-con unit plugged into it that's a far higher power draw. Also I don't know where you live and how good your country's electrical standards are, so I cant comment any further.
2.) Are the power bricks going to be in this cabinet as well.?