Step 1: Recall nutritional sources.
- Cod fish is rich in $\omega$-3 fatty acids such as EPA (Eicosapentaenoic acid) and DHA (Docosahexaenoic acid). Statement (A) is correct.
- Beetroot contains betalains but also contributes carotenoids including $\beta$-carotene, making statement (B) acceptable.
- Apples provide dietary fiber, vitamin C, and polyphenols but do not contain vitamin B$_{12}$, which is found only in animal-based foods like meat, fish, eggs, and dairy. Thus, statement (C) is not correct.
- Fresh sugarcane juice contains the enzyme polyphenol oxidase, which causes browning. Statement (D) is correct.
Step 2: Conclude.
The incorrect statement is that apples are a good source of vitamin B$_{12}$.
Final Answer:
\[
\boxed{\text{Apple is a good source of vitamin B$_{12}$ (Incorrect)}}
\]