It is correct, if you are speaking generally (not about a specific bird). "I don't cook duck" and "I don't cook ducks" would both work and be understood. There is a slight difference, though: "duck" would be talking about the food and "ducks" about the animal. "I don't cook a duck" sounds awkward because the article is usually dropped in English.
Changing a word to match gender only happens with nouns and adjectives. With verbs, you change the ending to match only the person who is performing the action. (gender excluded) For example: Yo hablo and ella/él/usted habla. It matches the speaker not the gender.
This was probably confusing but hope it helped a little!