Are there any methods to quantify H2O2 (hydrogen peroxide) which don't rely on horseradish peroxidase?
Amperometric detection of H2O2 can be performed without HRP, and gives high-resolution real-time readings.

You will find many different type of electrodes that are used in the literature to detect H2O2, including for instance MnO2-coated carbon paste microelectrodes or gold electrodes derivatized with cytochrome c.

Commercial electrodes also exists. For instance these from WPI are based on derivatized carbon fibers (although they do not specify with what, so I would not exclude HRP...)

Another option is that of using the H2O2-sensitive dye H2DCF (dihydro-dichlorofluorescein).