Question

is farmer a common noun or a proper noun

Answer 1

Proper nouns have capital letters at the beginning.

town, village, city = common nouns

Des Moines, New York, Riverside = proper nouns

What will you decide?

Answer 2

A farmer farmed
Farmer Jones farmed
See the difference?

Answer 3

I suspect the questioner is asking whether Farmer Brown farms, or farmer Brown farms.
See the difference?
If the farmer Mr Brown is generally known as Farmer Brown, the word farmer has been elevated to a nominative category unique to Mr Brown and thus ascends the rank to be regarded as proper, categorically speaking; thus, you can happily apply a capital.

Answer 4

is farmer a noun