Anyone can have an Agri registered quad or other vehicle, it’s not limited to ‘farm’ related ones - I live in the centre of town (albeit a rural one) and mine is Agri registered.
When it comes to road use the DVLA webpage is deliberately obtuse, which is why so many confuse vehicles registered as ‘limited use’ - which are limited to using public roads between different areas of land in the same occupation not exceeding 1.5km travelled - and Agri registration, which have unlimited road usage.
Here’s chapter and verse from the DVLA:
“If the vehicle is registered as an agricultural vehicle (quad), you are able to use it on the road as long as it is used
for purposes relating to agriculture; horticulture; forestry or activities falling within cutting grass verges; trees or hedges that border public roads. There is no restriction on distance.”
How you apply the ‘related purpose’ you are using the quad for at any given time is up you.