Calculating air flow velocity given volume per unit time.

Eddy currents will make this calculation suspect, but the face velocity at the furface of the filter is the Q (volume/time) divided by the filter area.

They should both be based on the same unit.

For instance the exhaust fan has a Q of 500 m^3/hr
and the filter area is 480/1000 m by 280/1000 m (0.1344m^2), the filter face velocity is 500/0.1344 (m^3/hr)/m^2=3270m/hr=0.908m/s=2.98 ft/s=2.03 mph

3 ft/s is a fair face velocity, but 15 m is a long piece of 6 inch duct which could well have a significant drag depending on duct roughness and number of ells and bends. Initially when the duct is clean it will be relatively smooth, but moisture and grease condensation and corrosion will rapidly increase the roughness.

I'd use 8 inch ducting or try to shorten the run.

Rap

The fan is not a positive displacement pump: instead it imparts a certain pressure increase to the air and the flow is determined by the downstream resistance. The quoted volumetric flowrate is valid for some backpressure that may or may not be contained in the fan specifications. From what you have described I expect the backpressure is likely a bit higher than was assumed in the specifications.