No. That subtly changes the meaning of the sentence.
In English, "our cat eats no sugar" really means "our cat does not eat any sugar".
In Spanish, the word "no" is not used this way. It's used to negate a verb.
If you wanted to emphasize that our cat doesn't eat any sugar in Spanish, it would be "Nuestra gata no come ninguno azúcar."
"Ninguno" used in a negative sentence means (roughly) "not any".
That sentence would still be best translated as "our cat doesn't eat any sugar", but "our cat eats no sugar" would also be acceptable.
Old comment so op probably knows this by now. But for anyone else wondering, comer is conjugated like this: Yo como, tú comes, él/ella/ustedes come, nosotros comemos, vosotros coméis, ellos/ellas/ustedes comen. La gata is third person singular(Él/ella,) not informal second person singular(tú.)
Because English requires an auxiliary verb with not; which one depends on the tense. No come=he does not eat, no comerá=he will not eat, no puede comer=he can not eat, no debe comer=he should not eat, no comí=he did not eat, no comería=he would not eat, no está comiendo= he is not eating.